%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE102+1 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n016.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 11:39:49 AM UTC 2026
% Result : Theorem 119.34s 15.63s
% Output : Proof 121.75s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05 % Problem : KLE102+1 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.08 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.20/0.46 % Computer : n016.cluster.edu
% 0.20/0.46 % Model : x86_64 x86_64
% 0.20/0.46 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.20/0.46 % Memory : 8046.5625MB
% 0.20/0.46 % OS : Linux 6.8.0-71-generic
% 0.20/0.46 % CPULimit : 300
% 0.20/0.46 % WCLimit : 300
% 0.20/0.46 % DateTime : Sun Sep 27 13:14:47 UTC 2026
% 0.20/0.46 % CPUTime :
% 0.20/0.46 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 119.34/15.63 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 119.34/15.63
% 119.34/15.63 % SZS status Theorem
% 119.34/15.63
% 120.94/15.88 % SZS output start Proof
% 120.94/15.88 Axiom 1 (domain4): domain(X) = antidomain(antidomain(X)).
% 120.94/15.88 Axiom 2 (complement): c(X) = antidomain(domain(X)).
% 120.94/15.88 Axiom 3 (right_annihilation): multiplication(X, zero) = zero.
% 120.94/15.88 Axiom 4 (multiplicative_right_identity): multiplication(X, one) = X.
% 120.94/15.88 Axiom 5 (left_annihilation): multiplication(zero, X) = zero.
% 120.94/15.88 Axiom 6 (multiplicative_left_identity): multiplication(one, X) = X.
% 120.94/15.88 Axiom 7 (domain1): multiplication(antidomain(X), X) = zero.
% 120.94/15.88 Axiom 8 (additive_idempotence): addition(X, X) = X.
% 120.94/15.88 Axiom 9 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 120.94/15.88 Axiom 10 (additive_identity): addition(X, zero) = X.
% 120.94/15.88 Axiom 11 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 120.94/15.88 Axiom 12 (codomain4): codomain(X) = coantidomain(coantidomain(X)).
% 120.94/15.88 Axiom 13 (codomain1): multiplication(X, coantidomain(X)) = zero.
% 120.94/15.88 Axiom 14 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 120.94/15.88 Axiom 15 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 120.94/15.88 Axiom 16 (forward_diamond): forward_diamond(X, Y) = domain(multiplication(X, domain(Y))).
% 120.94/15.88 Axiom 17 (forward_box): forward_box(X, Y) = c(forward_diamond(X, c(Y))).
% 120.94/15.88 Axiom 18 (backward_diamond): backward_diamond(X, Y) = codomain(multiplication(codomain(Y), X)).
% 120.94/15.88 Axiom 19 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 120.94/15.88 Axiom 20 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 120.94/15.88 Axiom 21 (codomain3): addition(coantidomain(coantidomain(X)), coantidomain(X)) = one.
% 120.94/15.88 Axiom 22 (goals): multiplication(domain(x1), backward_diamond(x0, domain(x2))) = zero.
% 120.94/15.88 Axiom 23 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 120.94/15.88 Axiom 24 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 120.94/15.88 Axiom 25 (domain2): addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))) = antidomain(multiplication(X, antidomain(antidomain(Y)))).
% 120.94/15.88 Axiom 26 (order_1): ifeq(leq(X, Y), true, addition(X, Y), Y) = Y.
% 120.94/15.88 Axiom 27 (order): ifeq2(addition(X, Y), Y, leq(X, Y), true) = true.
% 120.94/15.88 Axiom 28 (codomain2): addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(coantidomain(coantidomain(X)), Y))) = coantidomain(multiplication(coantidomain(coantidomain(X)), Y)).
% 120.94/15.88
% 120.94/15.88 Lemma 29: antidomain(one) = zero.
% 120.94/15.88 Proof:
% 120.94/15.88 antidomain(one)
% 120.94/15.88 = { by axiom 4 (multiplicative_right_identity) R->L }
% 120.94/15.88 multiplication(antidomain(one), one)
% 120.94/15.88 = { by axiom 7 (domain1) }
% 120.94/15.88 zero
% 120.94/15.88
% 120.94/15.88 Lemma 30: addition(zero, X) = X.
% 120.94/15.88 Proof:
% 120.94/15.88 addition(zero, X)
% 120.94/15.88 = { by axiom 9 (additive_commutativity) R->L }
% 120.94/15.88 addition(X, zero)
% 120.94/15.88 = { by axiom 10 (additive_identity) }
% 120.94/15.88 X
% 120.94/15.88
% 120.94/15.88 Lemma 31: addition(antidomain(X), antidomain(antidomain(X))) = one.
% 120.94/15.88 Proof:
% 120.94/15.88 addition(antidomain(X), antidomain(antidomain(X)))
% 120.94/15.88 = { by axiom 9 (additive_commutativity) R->L }
% 120.94/15.88 addition(antidomain(antidomain(X)), antidomain(X))
% 120.94/15.88 = { by axiom 11 (domain3) }
% 120.94/15.88 one
% 120.94/15.88
% 120.94/15.88 Lemma 32: antidomain(zero) = one.
% 120.94/15.88 Proof:
% 120.94/15.88 antidomain(zero)
% 120.94/15.88 = { by lemma 29 R->L }
% 120.94/15.88 antidomain(antidomain(one))
% 120.94/15.88 = { by lemma 30 R->L }
% 120.94/15.88 addition(zero, antidomain(antidomain(one)))
% 120.94/15.88 = { by lemma 29 R->L }
% 120.94/15.88 addition(antidomain(one), antidomain(antidomain(one)))
% 120.94/15.88 = { by lemma 31 }
% 120.94/15.88 one
% 120.94/15.88
% 120.94/15.88 Lemma 33: coantidomain(one) = zero.
% 120.94/15.88 Proof:
% 120.94/15.88 coantidomain(one)
% 120.94/15.88 = { by axiom 6 (multiplicative_left_identity) R->L }
% 120.94/15.88 multiplication(one, coantidomain(one))
% 120.94/15.88 = { by axiom 13 (codomain1) }
% 120.94/15.88 zero
% 120.94/15.88
% 120.94/15.88 Lemma 34: c(X) = antidomain(antidomain(antidomain(X))).
% 120.94/15.88 Proof:
% 120.94/15.88 c(X)
% 120.94/15.88 = { by axiom 2 (complement) }
% 120.94/15.88 antidomain(domain(X))
% 120.94/15.88 = { by axiom 1 (domain4) }
% 120.94/15.88 antidomain(antidomain(antidomain(X)))
% 120.94/15.88
% 120.94/15.88 Lemma 35: addition(domain(X), antidomain(X)) = one.
% 120.94/15.88 Proof:
% 120.94/15.88 addition(domain(X), antidomain(X))
% 120.94/15.88 = { by axiom 9 (additive_commutativity) R->L }
% 120.94/15.88 addition(antidomain(X), domain(X))
% 120.94/15.88 = { by axiom 1 (domain4) }
% 120.94/15.88 addition(antidomain(X), antidomain(antidomain(X)))
% 120.94/15.88 = { by lemma 31 }
% 120.94/15.88 one
% 120.94/15.88
% 120.94/15.88 Lemma 36: multiplication(antidomain(X), addition(X, Y)) = multiplication(antidomain(X), Y).
% 120.94/15.88 Proof:
% 120.94/15.88 multiplication(antidomain(X), addition(X, Y))
% 120.94/15.88 = { by axiom 9 (additive_commutativity) R->L }
% 120.94/15.88 multiplication(antidomain(X), addition(Y, X))
% 120.94/15.88 = { by axiom 23 (right_distributivity) }
% 120.94/15.88 addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X))
% 120.94/15.88 = { by axiom 7 (domain1) }
% 120.94/15.88 addition(multiplication(antidomain(X), Y), zero)
% 120.94/15.88 = { by axiom 10 (additive_identity) }
% 120.94/15.88 multiplication(antidomain(X), Y)
% 120.94/15.88
% 120.94/15.88 Lemma 37: multiplication(addition(X, antidomain(Y)), Y) = multiplication(X, Y).
% 120.94/15.88 Proof:
% 120.94/15.88 multiplication(addition(X, antidomain(Y)), Y)
% 120.94/15.88 = { by axiom 24 (left_distributivity) }
% 120.94/15.88 addition(multiplication(X, Y), multiplication(antidomain(Y), Y))
% 120.94/15.88 = { by axiom 7 (domain1) }
% 120.94/15.88 addition(multiplication(X, Y), zero)
% 120.94/15.88 = { by axiom 10 (additive_identity) }
% 120.94/15.88 multiplication(X, Y)
% 120.94/15.88
% 120.94/15.88 Lemma 38: multiplication(antidomain(antidomain(X)), X) = X.
% 120.94/15.88 Proof:
% 120.94/15.88 multiplication(antidomain(antidomain(X)), X)
% 120.94/15.88 = { by lemma 37 R->L }
% 120.94/15.88 multiplication(addition(antidomain(antidomain(X)), antidomain(X)), X)
% 120.94/15.88 = { by axiom 9 (additive_commutativity) }
% 120.94/15.88 multiplication(addition(antidomain(X), antidomain(antidomain(X))), X)
% 120.94/15.88 = { by lemma 31 }
% 120.94/15.88 multiplication(one, X)
% 120.94/15.88 = { by axiom 6 (multiplicative_left_identity) }
% 120.94/15.88 X
% 120.94/15.88
% 120.94/15.88 Lemma 39: antidomain(domain(X)) = antidomain(X).
% 120.94/15.88 Proof:
% 120.94/15.88 antidomain(domain(X))
% 120.94/15.88 = { by axiom 4 (multiplicative_right_identity) R->L }
% 120.94/15.88 multiplication(antidomain(domain(X)), one)
% 120.94/15.88 = { by lemma 35 R->L }
% 120.94/15.88 multiplication(antidomain(domain(X)), addition(domain(X), antidomain(X)))
% 120.94/15.88 = { by lemma 36 }
% 120.94/15.88 multiplication(antidomain(domain(X)), antidomain(X))
% 120.94/15.88 = { by axiom 1 (domain4) }
% 120.94/15.88 multiplication(antidomain(antidomain(antidomain(X))), antidomain(X))
% 120.94/15.88 = { by lemma 38 }
% 120.94/15.88 antidomain(X)
% 120.94/15.88
% 120.94/15.88 Lemma 40: addition(coantidomain(X), codomain(X)) = one.
% 120.94/15.88 Proof:
% 120.94/15.88 addition(coantidomain(X), codomain(X))
% 120.94/15.88 = { by axiom 9 (additive_commutativity) R->L }
% 120.94/15.88 addition(codomain(X), coantidomain(X))
% 120.94/15.88 = { by axiom 12 (codomain4) }
% 120.94/15.88 addition(coantidomain(coantidomain(X)), coantidomain(X))
% 120.94/15.88 = { by axiom 21 (codomain3) }
% 120.94/15.88 one
% 120.94/15.88
% 120.94/15.88 Lemma 41: addition(X, addition(X, Y)) = addition(X, Y).
% 120.94/15.88 Proof:
% 120.94/15.88 addition(X, addition(X, Y))
% 120.94/15.88 = { by axiom 15 (additive_associativity) }
% 120.94/15.88 addition(addition(X, X), Y)
% 120.94/15.88 = { by axiom 8 (additive_idempotence) }
% 120.94/15.88 addition(X, Y)
% 120.94/15.88
% 120.94/15.88 Lemma 42: addition(one, coantidomain(X)) = one.
% 120.94/15.88 Proof:
% 120.94/15.88 addition(one, coantidomain(X))
% 120.94/15.88 = { by axiom 9 (additive_commutativity) R->L }
% 120.94/15.88 addition(coantidomain(X), one)
% 120.94/15.88 = { by lemma 40 R->L }
% 120.94/15.88 addition(coantidomain(X), addition(coantidomain(X), codomain(X)))
% 120.94/15.88 = { by lemma 41 }
% 120.94/15.88 addition(coantidomain(X), codomain(X))
% 120.94/15.88 = { by lemma 40 }
% 120.94/15.88 one
% 120.94/15.88
% 120.94/15.88 Lemma 43: multiplication(addition(X, Y), coantidomain(Y)) = multiplication(X, coantidomain(Y)).
% 120.94/15.88 Proof:
% 120.94/15.88 multiplication(addition(X, Y), coantidomain(Y))
% 120.94/15.88 = { by axiom 24 (left_distributivity) }
% 120.94/15.88 addition(multiplication(X, coantidomain(Y)), multiplication(Y, coantidomain(Y)))
% 120.94/15.88 = { by axiom 13 (codomain1) }
% 120.94/15.88 addition(multiplication(X, coantidomain(Y)), zero)
% 120.94/15.88 = { by axiom 10 (additive_identity) }
% 120.94/15.88 multiplication(X, coantidomain(Y))
% 120.94/15.89
% 120.94/15.89 Lemma 44: codomain(coantidomain(X)) = coantidomain(codomain(X)).
% 120.94/15.89 Proof:
% 120.94/15.89 codomain(coantidomain(X))
% 120.94/15.89 = { by axiom 12 (codomain4) }
% 120.94/15.89 coantidomain(coantidomain(coantidomain(X)))
% 120.94/15.89 = { by axiom 12 (codomain4) R->L }
% 120.94/15.89 coantidomain(codomain(X))
% 120.94/15.89
% 120.94/15.89 Lemma 45: multiplication(X, addition(Y, coantidomain(X))) = multiplication(X, Y).
% 120.94/15.89 Proof:
% 120.94/15.89 multiplication(X, addition(Y, coantidomain(X)))
% 120.94/15.89 = { by axiom 23 (right_distributivity) }
% 120.94/15.89 addition(multiplication(X, Y), multiplication(X, coantidomain(X)))
% 120.94/15.89 = { by axiom 13 (codomain1) }
% 120.94/15.89 addition(multiplication(X, Y), zero)
% 120.94/15.89 = { by axiom 10 (additive_identity) }
% 120.94/15.89 multiplication(X, Y)
% 120.94/15.89
% 120.94/15.89 Lemma 46: multiplication(X, codomain(X)) = X.
% 120.94/15.89 Proof:
% 120.94/15.89 multiplication(X, codomain(X))
% 120.94/15.89 = { by lemma 45 R->L }
% 120.94/15.89 multiplication(X, addition(codomain(X), coantidomain(X)))
% 120.94/15.89 = { by axiom 9 (additive_commutativity) }
% 120.94/15.89 multiplication(X, addition(coantidomain(X), codomain(X)))
% 120.94/15.89 = { by lemma 40 }
% 120.94/15.89 multiplication(X, one)
% 120.94/15.89 = { by axiom 4 (multiplicative_right_identity) }
% 120.94/15.89 X
% 120.94/15.89
% 120.94/15.89 Lemma 47: coantidomain(codomain(X)) = coantidomain(X).
% 120.94/15.89 Proof:
% 120.94/15.89 coantidomain(codomain(X))
% 120.94/15.89 = { by axiom 6 (multiplicative_left_identity) R->L }
% 120.94/15.89 multiplication(one, coantidomain(codomain(X)))
% 120.94/15.89 = { by lemma 40 R->L }
% 120.94/15.89 multiplication(addition(coantidomain(X), codomain(X)), coantidomain(codomain(X)))
% 120.94/15.89 = { by lemma 43 }
% 120.94/15.89 multiplication(coantidomain(X), coantidomain(codomain(X)))
% 120.94/15.89 = { by lemma 44 R->L }
% 120.94/15.89 multiplication(coantidomain(X), codomain(coantidomain(X)))
% 120.94/15.89 = { by lemma 46 }
% 120.94/15.89 coantidomain(X)
% 120.94/15.89
% 120.94/15.89 Lemma 48: coantidomain(multiplication(codomain(X), Y)) = coantidomain(backward_diamond(Y, X)).
% 120.94/15.89 Proof:
% 120.94/15.89 coantidomain(multiplication(codomain(X), Y))
% 120.94/15.89 = { by lemma 47 R->L }
% 120.94/15.89 coantidomain(codomain(multiplication(codomain(X), Y)))
% 120.94/15.89 = { by axiom 18 (backward_diamond) R->L }
% 120.94/15.89 coantidomain(backward_diamond(Y, X))
% 120.94/15.89
% 120.94/15.89 Lemma 49: addition(codomain(X), coantidomain(codomain(X))) = one.
% 120.94/15.89 Proof:
% 120.94/15.89 addition(codomain(X), coantidomain(codomain(X)))
% 120.94/15.89 = { by axiom 12 (codomain4) }
% 120.94/15.89 addition(coantidomain(coantidomain(X)), coantidomain(codomain(X)))
% 120.94/15.89 = { by lemma 44 R->L }
% 120.94/15.89 addition(coantidomain(coantidomain(X)), codomain(coantidomain(X)))
% 120.94/15.89 = { by lemma 40 }
% 120.94/15.89 one
% 120.94/15.89
% 120.94/15.89 Lemma 50: multiplication(addition(X, coantidomain(Y)), codomain(Y)) = multiplication(X, codomain(Y)).
% 120.94/15.89 Proof:
% 120.94/15.89 multiplication(addition(X, coantidomain(Y)), codomain(Y))
% 120.94/15.89 = { by axiom 9 (additive_commutativity) R->L }
% 120.94/15.89 multiplication(addition(coantidomain(Y), X), codomain(Y))
% 120.94/15.89 = { by axiom 24 (left_distributivity) }
% 120.94/15.89 addition(multiplication(coantidomain(Y), codomain(Y)), multiplication(X, codomain(Y)))
% 120.94/15.89 = { by axiom 12 (codomain4) }
% 120.94/15.89 addition(multiplication(coantidomain(Y), coantidomain(coantidomain(Y))), multiplication(X, codomain(Y)))
% 120.94/15.89 = { by axiom 13 (codomain1) }
% 120.94/15.89 addition(zero, multiplication(X, codomain(Y)))
% 120.94/15.89 = { by lemma 30 }
% 120.94/15.89 multiplication(X, codomain(Y))
% 120.94/15.89
% 120.94/15.89 Lemma 51: addition(X, multiplication(X, Y)) = multiplication(X, addition(Y, one)).
% 120.94/15.89 Proof:
% 120.94/15.89 addition(X, multiplication(X, Y))
% 120.94/15.89 = { by axiom 4 (multiplicative_right_identity) R->L }
% 120.94/15.89 addition(multiplication(X, one), multiplication(X, Y))
% 120.94/15.89 = { by axiom 23 (right_distributivity) R->L }
% 120.94/15.89 multiplication(X, addition(one, Y))
% 120.94/15.89 = { by axiom 9 (additive_commutativity) }
% 120.94/15.89 multiplication(X, addition(Y, one))
% 120.94/15.89
% 120.94/15.89 Lemma 52: addition(multiplication(X, addition(one, Z)), Y) = addition(X, addition(Y, multiplication(X, Z))).
% 120.94/15.89 Proof:
% 120.94/15.89 addition(multiplication(X, addition(one, Z)), Y)
% 120.94/15.89 = { by axiom 9 (additive_commutativity) R->L }
% 120.94/15.89 addition(multiplication(X, addition(Z, one)), Y)
% 120.94/15.89 = { by lemma 51 R->L }
% 120.94/15.89 addition(addition(X, multiplication(X, Z)), Y)
% 120.94/15.89 = { by axiom 15 (additive_associativity) R->L }
% 120.94/15.89 addition(X, addition(multiplication(X, Z), Y))
% 120.94/15.89 = { by axiom 9 (additive_commutativity) }
% 120.94/15.89 addition(X, addition(Y, multiplication(X, Z)))
% 120.94/15.89
% 120.94/15.89 Lemma 53: addition(domain(X), antidomain(domain(X))) = one.
% 120.94/15.89 Proof:
% 120.94/15.89 addition(domain(X), antidomain(domain(X)))
% 120.94/15.89 = { by axiom 1 (domain4) }
% 120.94/15.89 addition(domain(X), antidomain(antidomain(antidomain(X))))
% 120.94/15.89 = { by axiom 1 (domain4) }
% 120.94/15.89 addition(antidomain(antidomain(X)), antidomain(antidomain(antidomain(X))))
% 120.94/15.89 = { by lemma 31 }
% 120.94/15.89 one
% 120.94/15.89
% 120.94/15.89 Lemma 54: addition(one, domain(X)) = one.
% 120.94/15.89 Proof:
% 120.94/15.89 addition(one, domain(X))
% 120.94/15.89 = { by axiom 9 (additive_commutativity) R->L }
% 120.94/15.89 addition(domain(X), one)
% 120.94/15.89 = { by lemma 53 R->L }
% 120.94/15.89 addition(domain(X), addition(domain(X), antidomain(domain(X))))
% 120.94/15.89 = { by lemma 41 }
% 120.94/15.89 addition(domain(X), antidomain(domain(X)))
% 120.94/15.89 = { by lemma 53 }
% 120.94/15.89 one
% 120.94/15.89
% 120.94/15.89 Lemma 55: addition(X, addition(Y, multiplication(X, domain(Z)))) = addition(X, Y).
% 120.94/15.89 Proof:
% 120.94/15.89 addition(X, addition(Y, multiplication(X, domain(Z))))
% 120.94/15.89 = { by lemma 52 R->L }
% 120.94/15.89 addition(multiplication(X, addition(one, domain(Z))), Y)
% 120.94/15.89 = { by lemma 54 }
% 120.94/15.89 addition(multiplication(X, one), Y)
% 120.94/15.89 = { by axiom 4 (multiplicative_right_identity) }
% 120.94/15.89 addition(X, Y)
% 120.94/15.89
% 120.94/15.89 Lemma 56: addition(X, addition(Y, multiplication(domain(Z), X))) = addition(X, Y).
% 120.94/15.89 Proof:
% 120.94/15.89 addition(X, addition(Y, multiplication(domain(Z), X)))
% 120.94/15.89 = { by axiom 9 (additive_commutativity) R->L }
% 120.94/15.89 addition(X, addition(multiplication(domain(Z), X), Y))
% 120.94/15.89 = { by axiom 15 (additive_associativity) }
% 120.94/15.89 addition(addition(X, multiplication(domain(Z), X)), Y)
% 120.94/15.89 = { by axiom 6 (multiplicative_left_identity) R->L }
% 120.94/15.89 addition(addition(multiplication(one, X), multiplication(domain(Z), X)), Y)
% 120.94/15.89 = { by axiom 24 (left_distributivity) R->L }
% 120.94/15.89 addition(multiplication(addition(one, domain(Z)), X), Y)
% 120.94/15.89 = { by lemma 54 }
% 120.94/15.89 addition(multiplication(one, X), Y)
% 120.94/15.89 = { by axiom 6 (multiplicative_left_identity) }
% 120.94/15.89 addition(X, Y)
% 120.94/15.89
% 120.94/15.89 Lemma 57: addition(Y, addition(X, Z)) = addition(X, addition(Y, Z)).
% 120.94/15.89 Proof:
% 120.94/15.89 addition(Y, addition(X, Z))
% 120.94/15.89 = { by axiom 9 (additive_commutativity) R->L }
% 120.94/15.89 addition(addition(X, Z), Y)
% 120.94/15.89 = { by axiom 15 (additive_associativity) R->L }
% 120.94/15.89 addition(X, addition(Z, Y))
% 120.94/15.89 = { by axiom 9 (additive_commutativity) }
% 120.94/15.89 addition(X, addition(Y, Z))
% 120.94/15.89
% 120.94/15.89 Lemma 58: coantidomain(antidomain(X)) = domain(X).
% 120.94/15.89 Proof:
% 120.94/15.89 coantidomain(antidomain(X))
% 120.94/15.89 = { by axiom 10 (additive_identity) R->L }
% 120.94/15.89 addition(coantidomain(antidomain(X)), zero)
% 120.94/15.89 = { by axiom 3 (right_annihilation) R->L }
% 120.94/15.89 addition(coantidomain(antidomain(X)), multiplication(domain(X), zero))
% 120.94/15.89 = { by axiom 3 (right_annihilation) R->L }
% 120.94/15.89 addition(coantidomain(antidomain(X)), multiplication(domain(X), multiplication(codomain(antidomain(X)), zero)))
% 120.94/15.89 = { by axiom 14 (multiplicative_associativity) }
% 120.94/15.89 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), zero))
% 120.94/15.89 = { by lemma 33 R->L }
% 120.94/15.89 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), coantidomain(one)))
% 120.94/15.89 = { by lemma 42 R->L }
% 120.94/15.89 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), coantidomain(addition(one, coantidomain(backward_diamond(multiplication(domain(X), codomain(antidomain(domain(X)))), antidomain(domain(X))))))))
% 120.94/15.89 = { by lemma 40 R->L }
% 120.94/15.89 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), coantidomain(addition(addition(coantidomain(one), codomain(one)), coantidomain(backward_diamond(multiplication(domain(X), codomain(antidomain(domain(X)))), antidomain(domain(X))))))))
% 120.94/15.89 = { by lemma 33 }
% 120.94/15.89 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), coantidomain(addition(addition(zero, codomain(one)), coantidomain(backward_diamond(multiplication(domain(X), codomain(antidomain(domain(X)))), antidomain(domain(X))))))))
% 120.94/15.89 = { by lemma 30 }
% 120.94/15.89 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), coantidomain(addition(codomain(one), coantidomain(backward_diamond(multiplication(domain(X), codomain(antidomain(domain(X)))), antidomain(domain(X))))))))
% 120.94/15.89 = { by axiom 12 (codomain4) }
% 120.94/15.89 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), coantidomain(addition(coantidomain(coantidomain(one)), coantidomain(backward_diamond(multiplication(domain(X), codomain(antidomain(domain(X)))), antidomain(domain(X))))))))
% 120.94/15.89 = { by lemma 33 }
% 120.94/15.89 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), coantidomain(addition(coantidomain(zero), coantidomain(backward_diamond(multiplication(domain(X), codomain(antidomain(domain(X)))), antidomain(domain(X))))))))
% 120.94/15.89 = { by axiom 5 (left_annihilation) R->L }
% 120.94/15.89 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), coantidomain(addition(coantidomain(multiplication(zero, codomain(antidomain(domain(X))))), coantidomain(backward_diamond(multiplication(domain(X), codomain(antidomain(domain(X)))), antidomain(domain(X))))))))
% 120.94/15.89 = { by axiom 7 (domain1) R->L }
% 120.94/15.89 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), coantidomain(addition(coantidomain(multiplication(multiplication(antidomain(domain(X)), domain(X)), codomain(antidomain(domain(X))))), coantidomain(backward_diamond(multiplication(domain(X), codomain(antidomain(domain(X)))), antidomain(domain(X))))))))
% 120.94/15.90 = { by axiom 14 (multiplicative_associativity) R->L }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), coantidomain(addition(coantidomain(multiplication(antidomain(domain(X)), multiplication(domain(X), codomain(antidomain(domain(X)))))), coantidomain(backward_diamond(multiplication(domain(X), codomain(antidomain(domain(X)))), antidomain(domain(X))))))))
% 120.94/15.90 = { by lemma 48 R->L }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), coantidomain(addition(coantidomain(multiplication(antidomain(domain(X)), multiplication(domain(X), codomain(antidomain(domain(X)))))), coantidomain(multiplication(codomain(antidomain(domain(X))), multiplication(domain(X), codomain(antidomain(domain(X))))))))))
% 120.94/15.90 = { by axiom 12 (codomain4) }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), coantidomain(addition(coantidomain(multiplication(antidomain(domain(X)), multiplication(domain(X), codomain(antidomain(domain(X)))))), coantidomain(multiplication(coantidomain(coantidomain(antidomain(domain(X)))), multiplication(domain(X), codomain(antidomain(domain(X))))))))))
% 120.94/15.90 = { by axiom 28 (codomain2) }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), coantidomain(coantidomain(multiplication(coantidomain(coantidomain(antidomain(domain(X)))), multiplication(domain(X), codomain(antidomain(domain(X)))))))))
% 120.94/15.90 = { by axiom 12 (codomain4) R->L }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), coantidomain(coantidomain(multiplication(codomain(antidomain(domain(X))), multiplication(domain(X), codomain(antidomain(domain(X)))))))))
% 120.94/15.90 = { by lemma 48 }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), coantidomain(coantidomain(backward_diamond(multiplication(domain(X), codomain(antidomain(domain(X)))), antidomain(domain(X)))))))
% 120.94/15.90 = { by axiom 12 (codomain4) R->L }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), codomain(backward_diamond(multiplication(domain(X), codomain(antidomain(domain(X)))), antidomain(domain(X))))))
% 120.94/15.90 = { by axiom 18 (backward_diamond) }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), codomain(codomain(multiplication(codomain(antidomain(domain(X))), multiplication(domain(X), codomain(antidomain(domain(X)))))))))
% 120.94/15.90 = { by axiom 12 (codomain4) }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), coantidomain(coantidomain(codomain(multiplication(codomain(antidomain(domain(X))), multiplication(domain(X), codomain(antidomain(domain(X))))))))))
% 120.94/15.90 = { by axiom 6 (multiplicative_left_identity) R->L }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), multiplication(one, coantidomain(coantidomain(codomain(multiplication(codomain(antidomain(domain(X))), multiplication(domain(X), codomain(antidomain(domain(X)))))))))))
% 120.94/15.90 = { by lemma 49 R->L }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), multiplication(addition(codomain(multiplication(codomain(antidomain(domain(X))), multiplication(domain(X), codomain(antidomain(domain(X)))))), coantidomain(codomain(multiplication(codomain(antidomain(domain(X))), multiplication(domain(X), codomain(antidomain(domain(X)))))))), coantidomain(coantidomain(codomain(multiplication(codomain(antidomain(domain(X))), multiplication(domain(X), codomain(antidomain(domain(X)))))))))))
% 120.94/15.90 = { by lemma 43 }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), multiplication(codomain(multiplication(codomain(antidomain(domain(X))), multiplication(domain(X), codomain(antidomain(domain(X)))))), coantidomain(coantidomain(codomain(multiplication(codomain(antidomain(domain(X))), multiplication(domain(X), codomain(antidomain(domain(X)))))))))))
% 120.94/15.90 = { by axiom 12 (codomain4) R->L }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), multiplication(codomain(multiplication(codomain(antidomain(domain(X))), multiplication(domain(X), codomain(antidomain(domain(X)))))), codomain(codomain(multiplication(codomain(antidomain(domain(X))), multiplication(domain(X), codomain(antidomain(domain(X))))))))))
% 120.94/15.90 = { by lemma 46 }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), codomain(multiplication(codomain(antidomain(domain(X))), multiplication(domain(X), codomain(antidomain(domain(X))))))))
% 120.94/15.90 = { by axiom 14 (multiplicative_associativity) }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), codomain(multiplication(multiplication(codomain(antidomain(domain(X))), domain(X)), codomain(antidomain(domain(X)))))))
% 120.94/15.90 = { by lemma 50 R->L }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), codomain(multiplication(addition(multiplication(codomain(antidomain(domain(X))), domain(X)), coantidomain(antidomain(domain(X)))), codomain(antidomain(domain(X)))))))
% 120.94/15.90 = { by axiom 9 (additive_commutativity) }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), codomain(multiplication(addition(coantidomain(antidomain(domain(X))), multiplication(codomain(antidomain(domain(X))), domain(X))), codomain(antidomain(domain(X)))))))
% 120.94/15.90 = { by lemma 55 R->L }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), codomain(multiplication(addition(coantidomain(antidomain(domain(X))), addition(multiplication(codomain(antidomain(domain(X))), domain(X)), multiplication(coantidomain(antidomain(domain(X))), domain(X)))), codomain(antidomain(domain(X)))))))
% 120.94/15.90 = { by axiom 24 (left_distributivity) R->L }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), codomain(multiplication(addition(coantidomain(antidomain(domain(X))), multiplication(addition(codomain(antidomain(domain(X))), coantidomain(antidomain(domain(X)))), domain(X))), codomain(antidomain(domain(X)))))))
% 120.94/15.90 = { by axiom 9 (additive_commutativity) }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), codomain(multiplication(addition(coantidomain(antidomain(domain(X))), multiplication(addition(coantidomain(antidomain(domain(X))), codomain(antidomain(domain(X)))), domain(X))), codomain(antidomain(domain(X)))))))
% 120.94/15.90 = { by lemma 40 }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), codomain(multiplication(addition(coantidomain(antidomain(domain(X))), multiplication(one, domain(X))), codomain(antidomain(domain(X)))))))
% 120.94/15.90 = { by axiom 6 (multiplicative_left_identity) }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), codomain(multiplication(addition(coantidomain(antidomain(domain(X))), domain(X)), codomain(antidomain(domain(X)))))))
% 120.94/15.90 = { by axiom 9 (additive_commutativity) }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), codomain(multiplication(addition(domain(X), coantidomain(antidomain(domain(X)))), codomain(antidomain(domain(X)))))))
% 120.94/15.90 = { by lemma 50 }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), codomain(multiplication(domain(X), codomain(antidomain(domain(X)))))))
% 120.94/15.90 = { by lemma 39 }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(multiplication(domain(X), codomain(antidomain(X))), codomain(multiplication(domain(X), codomain(antidomain(X))))))
% 120.94/15.90 = { by lemma 46 }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(domain(X), codomain(antidomain(X))))
% 120.94/15.90 = { by lemma 56 R->L }
% 120.94/15.90 addition(coantidomain(antidomain(X)), addition(multiplication(domain(X), codomain(antidomain(X))), multiplication(domain(X), coantidomain(antidomain(X)))))
% 120.94/15.90 = { by axiom 23 (right_distributivity) R->L }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(domain(X), addition(codomain(antidomain(X)), coantidomain(antidomain(X)))))
% 120.94/15.90 = { by axiom 9 (additive_commutativity) }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(domain(X), addition(coantidomain(antidomain(X)), codomain(antidomain(X)))))
% 120.94/15.90 = { by lemma 40 }
% 120.94/15.90 addition(coantidomain(antidomain(X)), multiplication(domain(X), one))
% 120.94/15.90 = { by axiom 4 (multiplicative_right_identity) }
% 120.94/15.90 addition(coantidomain(antidomain(X)), domain(X))
% 120.94/15.90 = { by axiom 9 (additive_commutativity) }
% 121.75/15.90 addition(domain(X), coantidomain(antidomain(X)))
% 121.75/15.90 = { by lemma 30 R->L }
% 121.75/15.90 addition(zero, addition(domain(X), coantidomain(antidomain(X))))
% 121.75/15.90 = { by lemma 57 }
% 121.75/15.90 addition(domain(X), addition(zero, coantidomain(antidomain(X))))
% 121.75/15.90 = { by lemma 39 R->L }
% 121.75/15.90 addition(domain(X), addition(zero, coantidomain(antidomain(domain(X)))))
% 121.75/15.90 = { by axiom 6 (multiplicative_left_identity) R->L }
% 121.75/15.90 addition(domain(X), addition(zero, multiplication(one, coantidomain(antidomain(domain(X))))))
% 121.75/15.90 = { by lemma 53 R->L }
% 121.75/15.90 addition(domain(X), addition(zero, multiplication(addition(domain(X), antidomain(domain(X))), coantidomain(antidomain(domain(X))))))
% 121.75/15.90 = { by lemma 43 }
% 121.75/15.90 addition(domain(X), addition(zero, multiplication(domain(X), coantidomain(antidomain(domain(X))))))
% 121.75/15.90 = { by lemma 39 }
% 121.75/15.90 addition(domain(X), addition(zero, multiplication(domain(X), coantidomain(antidomain(X)))))
% 121.75/15.90 = { by lemma 52 R->L }
% 121.75/15.90 addition(multiplication(domain(X), addition(one, coantidomain(antidomain(X)))), zero)
% 121.75/15.90 = { by lemma 42 }
% 121.75/15.90 addition(multiplication(domain(X), one), zero)
% 121.75/15.90 = { by axiom 4 (multiplicative_right_identity) }
% 121.75/15.90 addition(domain(X), zero)
% 121.75/15.90 = { by axiom 9 (additive_commutativity) }
% 121.75/15.90 addition(zero, domain(X))
% 121.75/15.90 = { by lemma 30 }
% 121.75/15.90 domain(X)
% 121.75/15.90
% 121.75/15.90 Lemma 59: multiplication(antidomain(X), addition(Y, X)) = multiplication(antidomain(X), Y).
% 121.75/15.90 Proof:
% 121.75/15.90 multiplication(antidomain(X), addition(Y, X))
% 121.75/15.90 = { by axiom 9 (additive_commutativity) R->L }
% 121.75/15.90 multiplication(antidomain(X), addition(X, Y))
% 121.75/15.90 = { by lemma 36 }
% 121.75/15.90 multiplication(antidomain(X), Y)
% 121.75/15.90
% 121.75/15.90 Lemma 60: multiplication(antidomain(addition(X, Y)), X) = zero.
% 121.75/15.90 Proof:
% 121.75/15.90 multiplication(antidomain(addition(X, Y)), X)
% 121.75/15.90 = { by lemma 59 R->L }
% 121.75/15.90 multiplication(antidomain(addition(X, Y)), addition(X, addition(X, Y)))
% 121.75/15.90 = { by lemma 41 }
% 121.75/15.90 multiplication(antidomain(addition(X, Y)), addition(X, Y))
% 121.75/15.90 = { by axiom 7 (domain1) }
% 121.75/15.90 zero
% 121.75/15.90
% 121.75/15.90 Lemma 61: antidomain(addition(X, one)) = zero.
% 121.75/15.90 Proof:
% 121.75/15.90 antidomain(addition(X, one))
% 121.75/15.90 = { by axiom 9 (additive_commutativity) R->L }
% 121.75/15.90 antidomain(addition(one, X))
% 121.75/15.90 = { by axiom 4 (multiplicative_right_identity) R->L }
% 121.75/15.90 multiplication(antidomain(addition(one, X)), one)
% 121.75/15.90 = { by lemma 60 }
% 121.75/15.90 zero
% 121.75/15.90
% 121.75/15.90 Lemma 62: antidomain(antidomain(antidomain(X))) = antidomain(X).
% 121.75/15.90 Proof:
% 121.75/15.90 antidomain(antidomain(antidomain(X)))
% 121.75/15.90 = { by axiom 4 (multiplicative_right_identity) R->L }
% 121.75/15.90 multiplication(antidomain(antidomain(antidomain(X))), one)
% 121.75/15.90 = { by lemma 31 R->L }
% 121.75/15.90 multiplication(antidomain(antidomain(antidomain(X))), addition(antidomain(X), antidomain(antidomain(X))))
% 121.75/15.90 = { by lemma 59 }
% 121.75/15.90 multiplication(antidomain(antidomain(antidomain(X))), antidomain(X))
% 121.75/15.90 = { by lemma 38 }
% 121.75/15.90 antidomain(X)
% 121.75/15.90
% 121.75/15.90 Lemma 63: antidomain(antidomain(multiplication(X, antidomain(antidomain(Y))))) = forward_diamond(X, Y).
% 121.75/15.90 Proof:
% 121.75/15.90 antidomain(antidomain(multiplication(X, antidomain(antidomain(Y)))))
% 121.75/15.90 = { by axiom 1 (domain4) R->L }
% 121.75/15.90 antidomain(antidomain(multiplication(X, domain(Y))))
% 121.75/15.90 = { by axiom 1 (domain4) R->L }
% 121.75/15.90 domain(multiplication(X, domain(Y)))
% 121.75/15.90 = { by axiom 16 (forward_diamond) R->L }
% 121.75/15.90 forward_diamond(X, Y)
% 121.75/15.90
% 121.75/15.90 Lemma 64: forward_diamond(X, antidomain(antidomain(Y))) = forward_diamond(X, Y).
% 121.75/15.90 Proof:
% 121.75/15.90 forward_diamond(X, antidomain(antidomain(Y)))
% 121.75/15.90 = { by lemma 63 R->L }
% 121.75/15.90 antidomain(antidomain(multiplication(X, antidomain(antidomain(antidomain(antidomain(Y)))))))
% 121.75/15.90 = { by lemma 62 }
% 121.75/15.90 antidomain(antidomain(multiplication(X, antidomain(antidomain(Y)))))
% 121.75/15.90 = { by lemma 63 }
% 121.75/15.90 forward_diamond(X, Y)
% 121.75/15.90
% 121.75/15.90 Lemma 65: antidomain(antidomain(antidomain(forward_diamond(X, antidomain(antidomain(antidomain(Y))))))) = forward_box(X, Y).
% 121.75/15.90 Proof:
% 121.75/15.90 antidomain(antidomain(antidomain(forward_diamond(X, antidomain(antidomain(antidomain(Y)))))))
% 121.75/15.90 = { by lemma 34 R->L }
% 121.75/15.90 antidomain(antidomain(antidomain(forward_diamond(X, c(Y)))))
% 121.75/15.90 = { by lemma 34 R->L }
% 121.75/15.90 c(forward_diamond(X, c(Y)))
% 121.75/15.90 = { by axiom 17 (forward_box) R->L }
% 121.75/15.90 forward_box(X, Y)
% 121.75/15.90
% 121.75/15.90 Lemma 66: antidomain(forward_diamond(X, antidomain(Y))) = forward_box(X, Y).
% 121.75/15.90 Proof:
% 121.75/15.90 antidomain(forward_diamond(X, antidomain(Y)))
% 121.75/15.90 = { by lemma 64 R->L }
% 121.75/15.90 antidomain(forward_diamond(X, antidomain(antidomain(antidomain(Y)))))
% 121.75/15.90 = { by lemma 62 R->L }
% 121.75/15.90 antidomain(antidomain(antidomain(forward_diamond(X, antidomain(antidomain(antidomain(Y)))))))
% 121.75/15.90 = { by lemma 65 }
% 121.75/15.90 forward_box(X, Y)
% 121.75/15.90
% 121.75/15.90 Lemma 67: antidomain(forward_diamond(X, Y)) = forward_box(X, antidomain(Y)).
% 121.75/15.90 Proof:
% 121.75/15.90 antidomain(forward_diamond(X, Y))
% 121.75/15.90 = { by lemma 64 R->L }
% 121.75/15.90 antidomain(forward_diamond(X, antidomain(antidomain(Y))))
% 121.75/15.90 = { by lemma 66 }
% 121.75/15.90 forward_box(X, antidomain(Y))
% 121.75/15.90
% 121.75/15.90 Lemma 68: addition(one, forward_diamond(X, Y)) = one.
% 121.75/15.90 Proof:
% 121.75/15.91 addition(one, forward_diamond(X, Y))
% 121.75/15.91 = { by lemma 63 R->L }
% 121.75/15.91 addition(one, antidomain(antidomain(multiplication(X, antidomain(antidomain(Y))))))
% 121.75/15.91 = { by axiom 9 (additive_commutativity) R->L }
% 121.75/15.91 addition(antidomain(antidomain(multiplication(X, antidomain(antidomain(Y))))), one)
% 121.75/15.91 = { by lemma 31 R->L }
% 121.75/15.91 addition(antidomain(antidomain(multiplication(X, antidomain(antidomain(Y))))), addition(antidomain(antidomain(multiplication(X, antidomain(antidomain(Y))))), antidomain(antidomain(antidomain(multiplication(X, antidomain(antidomain(Y))))))))
% 121.75/15.91 = { by lemma 41 }
% 121.75/15.91 addition(antidomain(antidomain(multiplication(X, antidomain(antidomain(Y))))), antidomain(antidomain(antidomain(multiplication(X, antidomain(antidomain(Y)))))))
% 121.75/15.91 = { by lemma 31 }
% 121.75/15.91 one
% 121.75/15.91
% 121.75/15.91 Lemma 69: addition(forward_diamond(X, Y), antidomain(forward_diamond(X, Y))) = one.
% 121.75/15.91 Proof:
% 121.75/15.91 addition(forward_diamond(X, Y), antidomain(forward_diamond(X, Y)))
% 121.75/15.91 = { by lemma 63 R->L }
% 121.75/15.91 addition(forward_diamond(X, Y), antidomain(antidomain(antidomain(multiplication(X, antidomain(antidomain(Y)))))))
% 121.75/15.91 = { by lemma 63 R->L }
% 121.75/15.91 addition(antidomain(antidomain(multiplication(X, antidomain(antidomain(Y))))), antidomain(antidomain(antidomain(multiplication(X, antidomain(antidomain(Y)))))))
% 121.75/15.91 = { by lemma 31 }
% 121.75/15.91 one
% 121.75/15.91
% 121.75/15.91 Lemma 70: antidomain(antidomain(forward_diamond(X, Y))) = forward_diamond(X, Y).
% 121.75/15.91 Proof:
% 121.75/15.91 antidomain(antidomain(forward_diamond(X, Y)))
% 121.75/15.91 = { by axiom 4 (multiplicative_right_identity) R->L }
% 121.75/15.91 multiplication(antidomain(antidomain(forward_diamond(X, Y))), one)
% 121.75/15.91 = { by lemma 69 R->L }
% 121.75/15.91 multiplication(antidomain(antidomain(forward_diamond(X, Y))), addition(forward_diamond(X, Y), antidomain(forward_diamond(X, Y))))
% 121.75/15.91 = { by lemma 59 }
% 121.75/15.91 multiplication(antidomain(antidomain(forward_diamond(X, Y))), forward_diamond(X, Y))
% 121.75/15.91 = { by lemma 38 }
% 121.75/15.91 forward_diamond(X, Y)
% 121.75/15.91
% 121.75/15.91 Lemma 71: coantidomain(backward_diamond(X, domain(Y))) = coantidomain(multiplication(domain(Y), X)).
% 121.75/15.91 Proof:
% 121.75/15.91 coantidomain(backward_diamond(X, domain(Y)))
% 121.75/15.91 = { by lemma 58 R->L }
% 121.75/15.91 coantidomain(backward_diamond(X, coantidomain(antidomain(Y))))
% 121.75/15.91 = { by axiom 18 (backward_diamond) }
% 121.75/15.91 coantidomain(codomain(multiplication(codomain(coantidomain(antidomain(Y))), X)))
% 121.75/15.91 = { by lemma 44 }
% 121.75/15.91 coantidomain(codomain(multiplication(coantidomain(codomain(antidomain(Y))), X)))
% 121.75/15.91 = { by lemma 47 }
% 121.75/15.91 coantidomain(multiplication(coantidomain(codomain(antidomain(Y))), X))
% 121.75/15.91 = { by lemma 47 }
% 121.75/15.91 coantidomain(multiplication(coantidomain(antidomain(Y)), X))
% 121.75/15.91 = { by lemma 58 }
% 121.75/15.91 coantidomain(multiplication(domain(Y), X))
% 121.75/15.91
% 121.75/15.91 Lemma 72: antidomain(multiplication(X, antidomain(antidomain(Y)))) = forward_box(X, antidomain(Y)).
% 121.75/15.91 Proof:
% 121.75/15.91 antidomain(multiplication(X, antidomain(antidomain(Y))))
% 121.75/15.91 = { by lemma 62 R->L }
% 121.75/15.91 antidomain(antidomain(antidomain(multiplication(X, antidomain(antidomain(Y))))))
% 121.75/15.91 = { by lemma 63 }
% 121.75/15.91 antidomain(forward_diamond(X, Y))
% 121.75/15.91 = { by lemma 67 }
% 121.75/15.91 forward_box(X, antidomain(Y))
% 121.75/15.91
% 121.75/15.91 Lemma 73: addition(multiplication(X, Y), multiplication(Z, Y)) = multiplication(addition(Z, X), Y).
% 121.75/15.91 Proof:
% 121.75/15.91 addition(multiplication(X, Y), multiplication(Z, Y))
% 121.75/15.91 = { by axiom 24 (left_distributivity) R->L }
% 121.75/15.91 multiplication(addition(X, Z), Y)
% 121.75/15.91 = { by axiom 9 (additive_commutativity) }
% 121.75/15.91 multiplication(addition(Z, X), Y)
% 121.75/15.91
% 121.75/15.91 Lemma 74: addition(X, multiplication(forward_diamond(Y, Z), X)) = X.
% 121.75/15.91 Proof:
% 121.75/15.91 addition(X, multiplication(forward_diamond(Y, Z), X))
% 121.75/15.91 = { by axiom 9 (additive_commutativity) R->L }
% 121.75/15.91 addition(multiplication(forward_diamond(Y, Z), X), X)
% 121.75/15.91 = { by axiom 6 (multiplicative_left_identity) R->L }
% 121.75/15.91 addition(multiplication(forward_diamond(Y, Z), X), multiplication(one, X))
% 121.75/15.91 = { by lemma 73 }
% 121.75/15.91 multiplication(addition(one, forward_diamond(Y, Z)), X)
% 121.75/15.91 = { by lemma 68 }
% 121.75/15.91 multiplication(one, X)
% 121.75/15.91 = { by axiom 6 (multiplicative_left_identity) }
% 121.75/15.91 X
% 121.75/15.91
% 121.75/15.91 Lemma 75: forward_box(addition(X, antidomain(Y)), antidomain(Y)) = forward_box(X, antidomain(Y)).
% 121.75/15.91 Proof:
% 121.75/15.91 forward_box(addition(X, antidomain(Y)), antidomain(Y))
% 121.75/15.91 = { by lemma 62 R->L }
% 121.75/15.91 forward_box(addition(X, antidomain(antidomain(antidomain(Y)))), antidomain(Y))
% 121.75/15.91 = { by lemma 72 R->L }
% 121.75/15.91 antidomain(multiplication(addition(X, antidomain(antidomain(antidomain(Y)))), antidomain(antidomain(Y))))
% 121.75/15.91 = { by lemma 37 }
% 121.75/15.91 antidomain(multiplication(X, antidomain(antidomain(Y))))
% 121.75/15.91 = { by lemma 72 }
% 121.75/15.91 forward_box(X, antidomain(Y))
% 121.75/15.91
% 121.75/15.91 Goal 1 (goals_1): multiplication(forward_diamond(x0, domain(x1)), domain(x2)) = zero.
% 121.75/15.91 Proof:
% 121.75/15.91 multiplication(forward_diamond(x0, domain(x1)), domain(x2))
% 121.75/15.91 = { by axiom 6 (multiplicative_left_identity) R->L }
% 121.75/15.91 multiplication(one, multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.91 = { by lemma 32 R->L }
% 121.75/15.91 multiplication(antidomain(zero), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.91 = { by lemma 60 R->L }
% 121.75/15.91 multiplication(antidomain(multiplication(antidomain(addition(antidomain(multiplication(domain(x2), multiplication(x0, domain(x1)))), forward_box(domain(x2), antidomain(multiplication(x0, domain(x1)))))), antidomain(multiplication(domain(x2), multiplication(x0, domain(x1)))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.91 = { by axiom 9 (additive_commutativity) }
% 121.75/15.91 multiplication(antidomain(multiplication(antidomain(addition(forward_box(domain(x2), antidomain(multiplication(x0, domain(x1)))), antidomain(multiplication(domain(x2), multiplication(x0, domain(x1)))))), antidomain(multiplication(domain(x2), multiplication(x0, domain(x1)))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.91 = { by lemma 62 R->L }
% 121.75/15.91 multiplication(antidomain(multiplication(antidomain(addition(forward_box(domain(x2), antidomain(multiplication(x0, domain(x1)))), antidomain(multiplication(domain(x2), multiplication(x0, domain(x1)))))), antidomain(antidomain(antidomain(multiplication(domain(x2), multiplication(x0, domain(x1)))))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.91 = { by lemma 72 }
% 121.75/15.91 multiplication(forward_box(antidomain(addition(forward_box(domain(x2), antidomain(multiplication(x0, domain(x1)))), antidomain(multiplication(domain(x2), multiplication(x0, domain(x1)))))), antidomain(antidomain(multiplication(domain(x2), multiplication(x0, domain(x1)))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.91 = { by lemma 66 R->L }
% 121.75/15.91 multiplication(antidomain(forward_diamond(antidomain(addition(forward_box(domain(x2), antidomain(multiplication(x0, domain(x1)))), antidomain(multiplication(domain(x2), multiplication(x0, domain(x1)))))), antidomain(antidomain(antidomain(multiplication(domain(x2), multiplication(x0, domain(x1)))))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.91 = { by lemma 62 R->L }
% 121.75/15.91 multiplication(antidomain(antidomain(antidomain(forward_diamond(antidomain(addition(forward_box(domain(x2), antidomain(multiplication(x0, domain(x1)))), antidomain(multiplication(domain(x2), multiplication(x0, domain(x1)))))), antidomain(antidomain(antidomain(multiplication(domain(x2), multiplication(x0, domain(x1)))))))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.91 = { by lemma 65 }
% 121.75/15.91 multiplication(forward_box(antidomain(addition(forward_box(domain(x2), antidomain(multiplication(x0, domain(x1)))), antidomain(multiplication(domain(x2), multiplication(x0, domain(x1)))))), multiplication(domain(x2), multiplication(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.91 = { by axiom 9 (additive_commutativity) R->L }
% 121.75/15.91 multiplication(forward_box(antidomain(addition(antidomain(multiplication(domain(x2), multiplication(x0, domain(x1)))), forward_box(domain(x2), antidomain(multiplication(x0, domain(x1)))))), multiplication(domain(x2), multiplication(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.91 = { by lemma 72 R->L }
% 121.75/15.91 multiplication(forward_box(antidomain(addition(antidomain(multiplication(domain(x2), multiplication(x0, domain(x1)))), antidomain(multiplication(domain(x2), antidomain(antidomain(multiplication(x0, domain(x1)))))))), multiplication(domain(x2), multiplication(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.91 = { by axiom 25 (domain2) }
% 121.75/15.91 multiplication(forward_box(antidomain(antidomain(multiplication(domain(x2), antidomain(antidomain(multiplication(x0, domain(x1))))))), multiplication(domain(x2), multiplication(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.91 = { by lemma 72 }
% 121.75/15.91 multiplication(forward_box(antidomain(forward_box(domain(x2), antidomain(multiplication(x0, domain(x1))))), multiplication(domain(x2), multiplication(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.91 = { by lemma 66 R->L }
% 121.75/15.91 multiplication(forward_box(antidomain(antidomain(forward_diamond(domain(x2), antidomain(antidomain(multiplication(x0, domain(x1))))))), multiplication(domain(x2), multiplication(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.91 = { by lemma 70 }
% 121.75/15.91 multiplication(forward_box(forward_diamond(domain(x2), antidomain(antidomain(multiplication(x0, domain(x1))))), multiplication(domain(x2), multiplication(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.91 = { by lemma 64 }
% 121.75/15.91 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), multiplication(domain(x2), multiplication(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.91 = { by axiom 14 (multiplicative_associativity) }
% 121.75/15.91 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), multiplication(multiplication(domain(x2), x0), domain(x1))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.91 = { by lemma 45 R->L }
% 121.75/15.91 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), multiplication(multiplication(domain(x2), x0), addition(domain(x1), coantidomain(multiplication(domain(x2), x0))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.91 = { by axiom 9 (additive_commutativity) R->L }
% 121.75/15.91 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), multiplication(multiplication(domain(x2), x0), addition(coantidomain(multiplication(domain(x2), x0)), domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.92 = { by lemma 71 R->L }
% 121.75/15.92 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), multiplication(multiplication(domain(x2), x0), addition(coantidomain(backward_diamond(x0, domain(x2))), domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.92 = { by axiom 19 (ifeq_axiom) R->L }
% 121.75/15.92 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), multiplication(multiplication(domain(x2), x0), ifeq(true, true, addition(coantidomain(backward_diamond(x0, domain(x2))), domain(x1)), coantidomain(backward_diamond(x0, domain(x2)))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.92 = { by axiom 27 (order) R->L }
% 121.75/15.92 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), multiplication(multiplication(domain(x2), x0), ifeq(ifeq2(addition(multiplication(domain(x1), coantidomain(backward_diamond(x0, domain(x2)))), addition(multiplication(domain(x1), coantidomain(backward_diamond(x0, domain(x2)))), addition(coantidomain(backward_diamond(x0, domain(x2))), coantidomain(backward_diamond(x0, domain(x2)))))), addition(multiplication(domain(x1), coantidomain(backward_diamond(x0, domain(x2)))), addition(coantidomain(backward_diamond(x0, domain(x2))), coantidomain(backward_diamond(x0, domain(x2))))), leq(multiplication(domain(x1), coantidomain(backward_diamond(x0, domain(x2)))), addition(multiplication(domain(x1), coantidomain(backward_diamond(x0, domain(x2)))), addition(coantidomain(backward_diamond(x0, domain(x2))), coantidomain(backward_diamond(x0, domain(x2)))))), true), true, addition(coantidomain(backward_diamond(x0, domain(x2))), domain(x1)), coantidomain(backward_diamond(x0, domain(x2)))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.92 = { by lemma 41 }
% 121.75/15.92 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), multiplication(multiplication(domain(x2), x0), ifeq(ifeq2(addition(multiplication(domain(x1), coantidomain(backward_diamond(x0, domain(x2)))), addition(coantidomain(backward_diamond(x0, domain(x2))), coantidomain(backward_diamond(x0, domain(x2))))), addition(multiplication(domain(x1), coantidomain(backward_diamond(x0, domain(x2)))), addition(coantidomain(backward_diamond(x0, domain(x2))), coantidomain(backward_diamond(x0, domain(x2))))), leq(multiplication(domain(x1), coantidomain(backward_diamond(x0, domain(x2)))), addition(multiplication(domain(x1), coantidomain(backward_diamond(x0, domain(x2)))), addition(coantidomain(backward_diamond(x0, domain(x2))), coantidomain(backward_diamond(x0, domain(x2)))))), true), true, addition(coantidomain(backward_diamond(x0, domain(x2))), domain(x1)), coantidomain(backward_diamond(x0, domain(x2)))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.92 = { by axiom 20 (ifeq_axiom) }
% 121.75/15.92 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), multiplication(multiplication(domain(x2), x0), ifeq(leq(multiplication(domain(x1), coantidomain(backward_diamond(x0, domain(x2)))), addition(multiplication(domain(x1), coantidomain(backward_diamond(x0, domain(x2)))), addition(coantidomain(backward_diamond(x0, domain(x2))), coantidomain(backward_diamond(x0, domain(x2)))))), true, addition(coantidomain(backward_diamond(x0, domain(x2))), domain(x1)), coantidomain(backward_diamond(x0, domain(x2)))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.92 = { by lemma 57 R->L }
% 121.75/15.92 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), multiplication(multiplication(domain(x2), x0), ifeq(leq(multiplication(domain(x1), coantidomain(backward_diamond(x0, domain(x2)))), addition(coantidomain(backward_diamond(x0, domain(x2))), addition(multiplication(domain(x1), coantidomain(backward_diamond(x0, domain(x2)))), coantidomain(backward_diamond(x0, domain(x2)))))), true, addition(coantidomain(backward_diamond(x0, domain(x2))), domain(x1)), coantidomain(backward_diamond(x0, domain(x2)))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.92 = { by axiom 9 (additive_commutativity) }
% 121.75/15.92 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), multiplication(multiplication(domain(x2), x0), ifeq(leq(multiplication(domain(x1), coantidomain(backward_diamond(x0, domain(x2)))), addition(coantidomain(backward_diamond(x0, domain(x2))), addition(coantidomain(backward_diamond(x0, domain(x2))), multiplication(domain(x1), coantidomain(backward_diamond(x0, domain(x2))))))), true, addition(coantidomain(backward_diamond(x0, domain(x2))), domain(x1)), coantidomain(backward_diamond(x0, domain(x2)))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.92 = { by lemma 56 }
% 121.75/15.93 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), multiplication(multiplication(domain(x2), x0), ifeq(leq(multiplication(domain(x1), coantidomain(backward_diamond(x0, domain(x2)))), addition(coantidomain(backward_diamond(x0, domain(x2))), coantidomain(backward_diamond(x0, domain(x2))))), true, addition(coantidomain(backward_diamond(x0, domain(x2))), domain(x1)), coantidomain(backward_diamond(x0, domain(x2)))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by axiom 8 (additive_idempotence) }
% 121.75/15.93 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), multiplication(multiplication(domain(x2), x0), ifeq(leq(multiplication(domain(x1), coantidomain(backward_diamond(x0, domain(x2)))), coantidomain(backward_diamond(x0, domain(x2)))), true, addition(coantidomain(backward_diamond(x0, domain(x2))), domain(x1)), coantidomain(backward_diamond(x0, domain(x2)))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by axiom 10 (additive_identity) R->L }
% 121.75/15.93 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), multiplication(multiplication(domain(x2), x0), ifeq(leq(addition(multiplication(domain(x1), coantidomain(backward_diamond(x0, domain(x2)))), zero), coantidomain(backward_diamond(x0, domain(x2)))), true, addition(coantidomain(backward_diamond(x0, domain(x2))), domain(x1)), coantidomain(backward_diamond(x0, domain(x2)))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by axiom 22 (goals) R->L }
% 121.75/15.93 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), multiplication(multiplication(domain(x2), x0), ifeq(leq(addition(multiplication(domain(x1), coantidomain(backward_diamond(x0, domain(x2)))), multiplication(domain(x1), backward_diamond(x0, domain(x2)))), coantidomain(backward_diamond(x0, domain(x2)))), true, addition(coantidomain(backward_diamond(x0, domain(x2))), domain(x1)), coantidomain(backward_diamond(x0, domain(x2)))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by axiom 23 (right_distributivity) R->L }
% 121.75/15.93 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), multiplication(multiplication(domain(x2), x0), ifeq(leq(multiplication(domain(x1), addition(coantidomain(backward_diamond(x0, domain(x2))), backward_diamond(x0, domain(x2)))), coantidomain(backward_diamond(x0, domain(x2)))), true, addition(coantidomain(backward_diamond(x0, domain(x2))), domain(x1)), coantidomain(backward_diamond(x0, domain(x2)))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by axiom 9 (additive_commutativity) }
% 121.75/15.93 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), multiplication(multiplication(domain(x2), x0), ifeq(leq(multiplication(domain(x1), addition(backward_diamond(x0, domain(x2)), coantidomain(backward_diamond(x0, domain(x2))))), coantidomain(backward_diamond(x0, domain(x2)))), true, addition(coantidomain(backward_diamond(x0, domain(x2))), domain(x1)), coantidomain(backward_diamond(x0, domain(x2)))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by axiom 18 (backward_diamond) }
% 121.75/15.93 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), multiplication(multiplication(domain(x2), x0), ifeq(leq(multiplication(domain(x1), addition(backward_diamond(x0, domain(x2)), coantidomain(codomain(multiplication(codomain(domain(x2)), x0))))), coantidomain(backward_diamond(x0, domain(x2)))), true, addition(coantidomain(backward_diamond(x0, domain(x2))), domain(x1)), coantidomain(backward_diamond(x0, domain(x2)))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by axiom 18 (backward_diamond) }
% 121.75/15.93 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), multiplication(multiplication(domain(x2), x0), ifeq(leq(multiplication(domain(x1), addition(codomain(multiplication(codomain(domain(x2)), x0)), coantidomain(codomain(multiplication(codomain(domain(x2)), x0))))), coantidomain(backward_diamond(x0, domain(x2)))), true, addition(coantidomain(backward_diamond(x0, domain(x2))), domain(x1)), coantidomain(backward_diamond(x0, domain(x2)))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by lemma 49 }
% 121.75/15.93 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), multiplication(multiplication(domain(x2), x0), ifeq(leq(multiplication(domain(x1), one), coantidomain(backward_diamond(x0, domain(x2)))), true, addition(coantidomain(backward_diamond(x0, domain(x2))), domain(x1)), coantidomain(backward_diamond(x0, domain(x2)))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by axiom 4 (multiplicative_right_identity) }
% 121.75/15.93 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), multiplication(multiplication(domain(x2), x0), ifeq(leq(domain(x1), coantidomain(backward_diamond(x0, domain(x2)))), true, addition(coantidomain(backward_diamond(x0, domain(x2))), domain(x1)), coantidomain(backward_diamond(x0, domain(x2)))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by axiom 9 (additive_commutativity) R->L }
% 121.75/15.93 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), multiplication(multiplication(domain(x2), x0), ifeq(leq(domain(x1), coantidomain(backward_diamond(x0, domain(x2)))), true, addition(domain(x1), coantidomain(backward_diamond(x0, domain(x2)))), coantidomain(backward_diamond(x0, domain(x2)))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by axiom 26 (order_1) }
% 121.75/15.93 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), multiplication(multiplication(domain(x2), x0), coantidomain(backward_diamond(x0, domain(x2))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by lemma 71 }
% 121.75/15.93 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), multiplication(multiplication(domain(x2), x0), coantidomain(multiplication(domain(x2), x0)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by axiom 13 (codomain1) }
% 121.75/15.93 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), zero), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by lemma 61 R->L }
% 121.75/15.93 multiplication(forward_box(forward_diamond(domain(x2), multiplication(x0, domain(x1))), antidomain(addition(X, one))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by lemma 67 R->L }
% 121.75/15.93 multiplication(antidomain(forward_diamond(forward_diamond(domain(x2), multiplication(x0, domain(x1))), addition(X, one))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by lemma 63 R->L }
% 121.75/15.93 multiplication(antidomain(antidomain(antidomain(multiplication(forward_diamond(domain(x2), multiplication(x0, domain(x1))), antidomain(antidomain(addition(X, one))))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by lemma 61 }
% 121.75/15.93 multiplication(antidomain(antidomain(antidomain(multiplication(forward_diamond(domain(x2), multiplication(x0, domain(x1))), antidomain(zero))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by lemma 32 }
% 121.75/15.93 multiplication(antidomain(antidomain(antidomain(multiplication(forward_diamond(domain(x2), multiplication(x0, domain(x1))), one)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by axiom 4 (multiplicative_right_identity) }
% 121.75/15.93 multiplication(antidomain(antidomain(antidomain(forward_diamond(domain(x2), multiplication(x0, domain(x1)))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by lemma 62 }
% 121.75/15.93 multiplication(antidomain(forward_diamond(domain(x2), multiplication(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by lemma 67 }
% 121.75/15.93 multiplication(forward_box(domain(x2), antidomain(multiplication(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by lemma 62 R->L }
% 121.75/15.93 multiplication(forward_box(domain(x2), antidomain(antidomain(antidomain(multiplication(x0, domain(x1)))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by axiom 1 (domain4) }
% 121.75/15.93 multiplication(forward_box(domain(x2), antidomain(antidomain(antidomain(multiplication(x0, antidomain(antidomain(x1))))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by lemma 63 }
% 121.75/15.93 multiplication(forward_box(domain(x2), antidomain(forward_diamond(x0, x1))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by lemma 67 }
% 121.75/15.93 multiplication(forward_box(domain(x2), forward_box(x0, antidomain(x1))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by lemma 39 R->L }
% 121.75/15.93 multiplication(forward_box(domain(x2), forward_box(x0, antidomain(domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by lemma 67 R->L }
% 121.75/15.93 multiplication(forward_box(domain(x2), antidomain(forward_diamond(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by lemma 75 R->L }
% 121.75/15.93 multiplication(forward_box(addition(domain(x2), antidomain(forward_diamond(x0, domain(x1)))), antidomain(forward_diamond(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by lemma 74 R->L }
% 121.75/15.93 multiplication(forward_box(addition(addition(domain(x2), multiplication(forward_diamond(x0, domain(x1)), domain(x2))), antidomain(forward_diamond(x0, domain(x1)))), antidomain(forward_diamond(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by axiom 15 (additive_associativity) R->L }
% 121.75/15.93 multiplication(forward_box(addition(domain(x2), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), antidomain(forward_diamond(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by axiom 19 (ifeq_axiom) R->L }
% 121.75/15.93 multiplication(forward_box(ifeq(true, true, addition(domain(x2), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), antidomain(forward_diamond(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by axiom 27 (order) R->L }
% 121.75/15.93 multiplication(forward_box(ifeq(ifeq2(addition(multiplication(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), addition(multiplication(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), addition(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))))), addition(multiplication(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), addition(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))))), leq(multiplication(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), addition(multiplication(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), addition(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))))), true), true, addition(domain(x2), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), antidomain(forward_diamond(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by lemma 41 }
% 121.75/15.93 multiplication(forward_box(ifeq(ifeq2(addition(multiplication(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), addition(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))))), addition(multiplication(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), addition(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))))), leq(multiplication(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), addition(multiplication(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), addition(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))))), true), true, addition(domain(x2), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), antidomain(forward_diamond(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by axiom 20 (ifeq_axiom) }
% 121.75/15.93 multiplication(forward_box(ifeq(leq(multiplication(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), addition(multiplication(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), addition(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))))), true, addition(domain(x2), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), antidomain(forward_diamond(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.93 = { by lemma 57 R->L }
% 121.75/15.94 multiplication(forward_box(ifeq(leq(multiplication(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), addition(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), addition(multiplication(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))))), true, addition(domain(x2), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), antidomain(forward_diamond(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by axiom 9 (additive_commutativity) }
% 121.75/15.94 multiplication(forward_box(ifeq(leq(multiplication(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), addition(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), addition(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), multiplication(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), domain(x2))))), true, addition(domain(x2), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), antidomain(forward_diamond(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by lemma 55 }
% 121.75/15.94 multiplication(forward_box(ifeq(leq(multiplication(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), addition(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))))), true, addition(domain(x2), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), antidomain(forward_diamond(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by axiom 8 (additive_idempotence) }
% 121.75/15.94 multiplication(forward_box(ifeq(leq(multiplication(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), true, addition(domain(x2), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), antidomain(forward_diamond(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by axiom 24 (left_distributivity) }
% 121.75/15.94 multiplication(forward_box(ifeq(leq(addition(multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), domain(x2)), multiplication(antidomain(forward_diamond(x0, domain(x1))), domain(x2))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), true, addition(domain(x2), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), antidomain(forward_diamond(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by axiom 14 (multiplicative_associativity) R->L }
% 121.75/15.94 multiplication(forward_box(ifeq(leq(addition(multiplication(forward_diamond(x0, domain(x1)), multiplication(domain(x2), domain(x2))), multiplication(antidomain(forward_diamond(x0, domain(x1))), domain(x2))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), true, addition(domain(x2), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), antidomain(forward_diamond(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by lemma 37 R->L }
% 121.75/15.94 multiplication(forward_box(ifeq(leq(addition(multiplication(forward_diamond(x0, domain(x1)), multiplication(addition(domain(x2), antidomain(domain(x2))), domain(x2))), multiplication(antidomain(forward_diamond(x0, domain(x1))), domain(x2))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), true, addition(domain(x2), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), antidomain(forward_diamond(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by lemma 53 }
% 121.75/15.94 multiplication(forward_box(ifeq(leq(addition(multiplication(forward_diamond(x0, domain(x1)), multiplication(one, domain(x2))), multiplication(antidomain(forward_diamond(x0, domain(x1))), domain(x2))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), true, addition(domain(x2), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), antidomain(forward_diamond(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by axiom 6 (multiplicative_left_identity) }
% 121.75/15.94 multiplication(forward_box(ifeq(leq(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), multiplication(antidomain(forward_diamond(x0, domain(x1))), domain(x2))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), true, addition(domain(x2), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), antidomain(forward_diamond(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by lemma 73 }
% 121.75/15.94 multiplication(forward_box(ifeq(leq(multiplication(addition(antidomain(forward_diamond(x0, domain(x1))), forward_diamond(x0, domain(x1))), domain(x2)), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), true, addition(domain(x2), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), antidomain(forward_diamond(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by axiom 9 (additive_commutativity) }
% 121.75/15.94 multiplication(forward_box(ifeq(leq(multiplication(addition(forward_diamond(x0, domain(x1)), antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), true, addition(domain(x2), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), antidomain(forward_diamond(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by lemma 69 }
% 121.75/15.94 multiplication(forward_box(ifeq(leq(multiplication(one, domain(x2)), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), true, addition(domain(x2), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), antidomain(forward_diamond(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by axiom 6 (multiplicative_left_identity) }
% 121.75/15.94 multiplication(forward_box(ifeq(leq(domain(x2), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), true, addition(domain(x2), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1))))), antidomain(forward_diamond(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by axiom 26 (order_1) }
% 121.75/15.94 multiplication(forward_box(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), antidomain(forward_diamond(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by lemma 75 }
% 121.75/15.94 multiplication(forward_box(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(forward_diamond(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by lemma 67 R->L }
% 121.75/15.94 multiplication(antidomain(forward_diamond(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), forward_diamond(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by lemma 63 R->L }
% 121.75/15.94 multiplication(antidomain(antidomain(antidomain(multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), antidomain(antidomain(forward_diamond(x0, domain(x1)))))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by lemma 70 }
% 121.75/15.94 multiplication(antidomain(antidomain(antidomain(multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), forward_diamond(x0, domain(x1)))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by lemma 62 }
% 121.75/15.94 multiplication(antidomain(multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), forward_diamond(x0, domain(x1)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by axiom 4 (multiplicative_right_identity) R->L }
% 121.75/15.94 multiplication(antidomain(multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), multiplication(forward_diamond(x0, domain(x1)), one))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by lemma 54 R->L }
% 121.75/15.94 multiplication(antidomain(multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), multiplication(forward_diamond(x0, domain(x1)), addition(one, domain(x2))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by axiom 9 (additive_commutativity) R->L }
% 121.75/15.94 multiplication(antidomain(multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), multiplication(forward_diamond(x0, domain(x1)), addition(domain(x2), one)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by axiom 23 (right_distributivity) }
% 121.75/15.94 multiplication(antidomain(multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), multiplication(forward_diamond(x0, domain(x1)), one)))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by axiom 4 (multiplicative_right_identity) }
% 121.75/15.94 multiplication(antidomain(multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), forward_diamond(x0, domain(x1))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by axiom 23 (right_distributivity) }
% 121.75/15.94 multiplication(antidomain(addition(multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), multiplication(forward_diamond(x0, domain(x1)), domain(x2))), multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), forward_diamond(x0, domain(x1))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by axiom 14 (multiplicative_associativity) R->L }
% 121.75/15.94 multiplication(antidomain(addition(multiplication(forward_diamond(x0, domain(x1)), multiplication(domain(x2), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), forward_diamond(x0, domain(x1))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by lemma 30 R->L }
% 121.75/15.94 multiplication(antidomain(addition(multiplication(forward_diamond(x0, domain(x1)), addition(zero, multiplication(domain(x2), multiplication(forward_diamond(x0, domain(x1)), domain(x2))))), multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), forward_diamond(x0, domain(x1))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.94 = { by axiom 7 (domain1) R->L }
% 121.75/15.94 multiplication(antidomain(addition(multiplication(forward_diamond(x0, domain(x1)), addition(multiplication(antidomain(domain(x2)), domain(x2)), multiplication(domain(x2), multiplication(forward_diamond(x0, domain(x1)), domain(x2))))), multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), forward_diamond(x0, domain(x1))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.95 = { by lemma 74 R->L }
% 121.75/15.95 multiplication(antidomain(addition(multiplication(forward_diamond(x0, domain(x1)), addition(multiplication(antidomain(domain(x2)), addition(domain(x2), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), multiplication(forward_diamond(x0, domain(x1)), domain(x2))))), multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), forward_diamond(x0, domain(x1))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.95 = { by lemma 36 }
% 121.75/15.95 multiplication(antidomain(addition(multiplication(forward_diamond(x0, domain(x1)), addition(multiplication(antidomain(domain(x2)), multiplication(forward_diamond(x0, domain(x1)), domain(x2))), multiplication(domain(x2), multiplication(forward_diamond(x0, domain(x1)), domain(x2))))), multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), forward_diamond(x0, domain(x1))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.95 = { by axiom 24 (left_distributivity) R->L }
% 121.75/15.95 multiplication(antidomain(addition(multiplication(forward_diamond(x0, domain(x1)), multiplication(addition(antidomain(domain(x2)), domain(x2)), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), forward_diamond(x0, domain(x1))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.95 = { by lemma 39 }
% 121.75/15.95 multiplication(antidomain(addition(multiplication(forward_diamond(x0, domain(x1)), multiplication(addition(antidomain(x2), domain(x2)), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), forward_diamond(x0, domain(x1))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.95 = { by axiom 9 (additive_commutativity) }
% 121.75/15.95 multiplication(antidomain(addition(multiplication(forward_diamond(x0, domain(x1)), multiplication(addition(domain(x2), antidomain(x2)), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), forward_diamond(x0, domain(x1))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.95 = { by lemma 35 }
% 121.75/15.95 multiplication(antidomain(addition(multiplication(forward_diamond(x0, domain(x1)), multiplication(one, multiplication(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), forward_diamond(x0, domain(x1))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.95 = { by axiom 6 (multiplicative_left_identity) }
% 121.75/15.95 multiplication(antidomain(addition(multiplication(forward_diamond(x0, domain(x1)), multiplication(forward_diamond(x0, domain(x1)), domain(x2))), multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), forward_diamond(x0, domain(x1))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.95 = { by axiom 14 (multiplicative_associativity) }
% 121.75/15.95 multiplication(antidomain(addition(multiplication(multiplication(forward_diamond(x0, domain(x1)), forward_diamond(x0, domain(x1))), domain(x2)), multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), forward_diamond(x0, domain(x1))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.95 = { by lemma 37 R->L }
% 121.75/15.95 multiplication(antidomain(addition(multiplication(multiplication(addition(forward_diamond(x0, domain(x1)), antidomain(forward_diamond(x0, domain(x1)))), forward_diamond(x0, domain(x1))), domain(x2)), multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), forward_diamond(x0, domain(x1))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.95 = { by lemma 69 }
% 121.75/15.95 multiplication(antidomain(addition(multiplication(multiplication(one, forward_diamond(x0, domain(x1))), domain(x2)), multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), forward_diamond(x0, domain(x1))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.95 = { by axiom 6 (multiplicative_left_identity) }
% 121.75/15.95 multiplication(antidomain(addition(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), forward_diamond(x0, domain(x1))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.95 = { by lemma 51 }
% 121.75/15.95 multiplication(antidomain(multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), addition(forward_diamond(x0, domain(x1)), one))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.95 = { by axiom 9 (additive_commutativity) }
% 121.75/15.95 multiplication(antidomain(multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), addition(one, forward_diamond(x0, domain(x1))))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.95 = { by lemma 68 }
% 121.75/15.95 multiplication(antidomain(multiplication(multiplication(forward_diamond(x0, domain(x1)), domain(x2)), one)), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.95 = { by axiom 4 (multiplicative_right_identity) }
% 121.75/15.95 multiplication(antidomain(multiplication(forward_diamond(x0, domain(x1)), domain(x2))), multiplication(forward_diamond(x0, domain(x1)), domain(x2)))
% 121.75/15.95 = { by axiom 7 (domain1) }
% 121.75/15.95 zero
% 121.75/15.95 % SZS output end Proof
% 121.75/15.95
% 121.75/15.95 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------