%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE112+1 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n003.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 11:39:50 AM UTC 2026
% Result : Timeout 293.38s 37.54s
% Output : None
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.04 % Problem : KLE112+1 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.07 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.20/0.45 % Computer : n003.cluster.edu
% 0.20/0.45 % Model : x86_64 x86_64
% 0.20/0.45 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.20/0.45 % Memory : 8046.5625MB
% 0.20/0.45 % OS : Linux 6.8.0-71-generic
% 0.20/0.45 % CPULimit : 300
% 0.20/0.45 % WCLimit : 300
% 0.20/0.45 % DateTime : Sun Sep 27 13:12:55 UTC 2026
% 0.20/0.46 % CPUTime :
% 0.20/0.46 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 293.38/37.54 Command-line arguments: --stitch /export/starexec/sandbox/solver/bin/stitch --hint-skel-cost 0 --hint-skel-factor 0.5
% 293.38/37.54
% 293.38/37.54 % SZS status Theorem
% 293.38/37.54
% 294.22/37.64 % SZS output start Proof
% 294.22/37.64 Axiom 1 (domain4): domain(X) = antidomain(antidomain(X)).
% 294.22/37.64 Axiom 2 (complement): c(X) = antidomain(domain(X)).
% 294.22/37.64 Axiom 3 (codomain4): codomain(X) = coantidomain(coantidomain(X)).
% 294.22/37.64 Axiom 4 (additive_idempotence): addition(X, X) = X.
% 294.22/37.64 Axiom 5 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 294.22/37.64 Axiom 6 (additive_identity): addition(X, zero) = X.
% 294.22/37.64 Axiom 7 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 294.22/37.64 Axiom 8 (multiplicative_right_identity): multiplication(X, one) = X.
% 294.22/37.64 Axiom 9 (right_annihilation): multiplication(X, zero) = zero.
% 294.22/37.64 Axiom 10 (multiplicative_left_identity): multiplication(one, X) = X.
% 294.22/37.64 Axiom 11 (domain1): multiplication(antidomain(X), X) = zero.
% 294.22/37.64 Axiom 12 (codomain1): multiplication(X, coantidomain(X)) = zero.
% 294.22/37.64 Axiom 13 (domain_difference): domain_difference(X, Y) = multiplication(domain(X), antidomain(Y)).
% 294.22/37.64 Axiom 14 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 294.22/37.64 Axiom 15 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 294.22/37.64 Axiom 16 (backward_diamond): backward_diamond(X, Y) = codomain(multiplication(codomain(Y), X)).
% 294.22/37.64 Axiom 17 (forward_box): forward_box(X, Y) = c(forward_diamond(X, c(Y))).
% 294.22/37.64 Axiom 18 (forward_diamond): forward_diamond(X, Y) = domain(multiplication(X, domain(Y))).
% 294.22/37.64 Axiom 19 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 294.22/37.65 Axiom 20 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 294.22/37.65 Axiom 21 (codomain3): addition(coantidomain(coantidomain(X)), coantidomain(X)) = one.
% 294.22/37.65 Axiom 22 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 294.22/37.65 Axiom 23 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 294.22/37.65 Axiom 24 (domain2): addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))) = antidomain(multiplication(X, antidomain(antidomain(Y)))).
% 294.22/37.65 Axiom 25 (order_1): ifeq(leq(X, Y), true, addition(X, Y), Y) = Y.
% 294.22/37.65 Axiom 26 (order): ifeq2(addition(X, Y), Y, leq(X, Y), true) = true.
% 294.22/37.65 Axiom 27 (codomain2): addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(coantidomain(coantidomain(X)), Y))) = coantidomain(multiplication(coantidomain(coantidomain(X)), Y)).
% 294.22/37.65
% 294.22/37.65 Lemma 28: antidomain(one) = zero.
% 294.22/37.65 Proof:
% 294.22/37.65 antidomain(one)
% 294.22/37.65 = { by axiom 8 (multiplicative_right_identity) R->L }
% 294.22/37.65 multiplication(antidomain(one), one)
% 294.22/37.65 = { by axiom 11 (domain1) }
% 294.22/37.65 zero
% 294.22/37.65
% 294.22/37.65 Lemma 29: addition(zero, X) = X.
% 294.22/37.65 Proof:
% 294.22/37.65 addition(zero, X)
% 294.22/37.65 = { by axiom 5 (additive_commutativity) R->L }
% 294.22/37.65 addition(X, zero)
% 294.22/37.65 = { by axiom 6 (additive_identity) }
% 294.22/37.65 X
% 294.22/37.65
% 294.22/37.65 Lemma 30: addition(antidomain(X), antidomain(antidomain(X))) = one.
% 294.22/37.65 Proof:
% 294.22/37.65 addition(antidomain(X), antidomain(antidomain(X)))
% 294.22/37.65 = { by axiom 5 (additive_commutativity) R->L }
% 294.22/37.65 addition(antidomain(antidomain(X)), antidomain(X))
% 294.22/37.65 = { by axiom 7 (domain3) }
% 294.22/37.65 one
% 294.22/37.65
% 294.22/37.65 Lemma 31: antidomain(zero) = one.
% 294.22/37.65 Proof:
% 294.22/37.65 antidomain(zero)
% 294.22/37.65 = { by lemma 28 R->L }
% 294.22/37.65 antidomain(antidomain(one))
% 294.22/37.65 = { by lemma 29 R->L }
% 294.22/37.65 addition(zero, antidomain(antidomain(one)))
% 294.22/37.65 = { by lemma 28 R->L }
% 294.22/37.65 addition(antidomain(one), antidomain(antidomain(one)))
% 294.22/37.65 = { by lemma 30 }
% 294.22/37.65 one
% 294.22/37.65
% 294.22/37.65 Lemma 32: coantidomain(one) = zero.
% 294.22/37.65 Proof:
% 294.22/37.65 coantidomain(one)
% 294.22/37.65 = { by axiom 10 (multiplicative_left_identity) R->L }
% 294.22/37.65 multiplication(one, coantidomain(one))
% 294.22/37.65 = { by axiom 12 (codomain1) }
% 294.22/37.65 zero
% 294.22/37.65
% 294.22/37.65 Lemma 33: addition(codomain(X), coantidomain(X)) = one.
% 294.22/37.65 Proof:
% 294.22/37.65 addition(codomain(X), coantidomain(X))
% 294.22/37.65 = { by axiom 3 (codomain4) }
% 294.22/37.65 addition(coantidomain(coantidomain(X)), coantidomain(X))
% 294.22/37.65 = { by axiom 21 (codomain3) }
% 294.22/37.65 one
% 294.22/37.65
% 294.22/37.65 Lemma 34: codomain(one) = one.
% 294.22/37.65 Proof:
% 294.22/37.65 codomain(one)
% 294.22/37.65 = { by axiom 6 (additive_identity) R->L }
% 294.22/37.65 addition(codomain(one), zero)
% 294.22/37.65 = { by lemma 32 R->L }
% 294.22/37.65 addition(codomain(one), coantidomain(one))
% 294.22/37.65 = { by lemma 33 }
% 294.22/37.65 one
% 294.22/37.65
% 294.22/37.65 Lemma 35: c(X) = antidomain(antidomain(antidomain(X))).
% 294.22/37.65 Proof:
% 294.22/37.65 c(X)
% 294.22/37.65 = { by axiom 2 (complement) }
% 294.22/37.65 antidomain(domain(X))
% 294.22/37.65 = { by axiom 1 (domain4) }
% 294.22/37.65 antidomain(antidomain(antidomain(X)))
% 294.22/37.65
% 294.22/37.65 Lemma 36: coantidomain(codomain(X)) = codomain(coantidomain(X)).
% 294.22/37.65 Proof:
% 294.22/37.65 coantidomain(codomain(X))
% 294.22/37.65 = { by axiom 3 (codomain4) }
% 294.22/37.65 coantidomain(coantidomain(coantidomain(X)))
% 294.22/37.65 = { by axiom 3 (codomain4) R->L }
% 294.22/37.65 codomain(coantidomain(X))
% 294.22/37.65
% 294.22/37.65 Lemma 37: multiplication(addition(X, Y), coantidomain(X)) = multiplication(Y, coantidomain(X)).
% 294.22/37.65 Proof:
% 294.22/37.65 multiplication(addition(X, Y), coantidomain(X))
% 294.22/37.65 = { by axiom 5 (additive_commutativity) R->L }
% 294.22/37.65 multiplication(addition(Y, X), coantidomain(X))
% 294.22/37.65 = { by axiom 23 (left_distributivity) }
% 294.22/37.65 addition(multiplication(Y, coantidomain(X)), multiplication(X, coantidomain(X)))
% 294.22/37.65 = { by axiom 12 (codomain1) }
% 294.22/37.65 addition(multiplication(Y, coantidomain(X)), zero)
% 294.22/37.65 = { by axiom 6 (additive_identity) }
% 294.22/37.65 multiplication(Y, coantidomain(X))
% 294.22/37.65
% 294.22/37.65 Lemma 38: multiplication(X, addition(Y, coantidomain(X))) = multiplication(X, Y).
% 294.22/37.65 Proof:
% 294.22/37.65 multiplication(X, addition(Y, coantidomain(X)))
% 294.22/37.65 = { by axiom 22 (right_distributivity) }
% 294.22/37.65 addition(multiplication(X, Y), multiplication(X, coantidomain(X)))
% 294.22/37.65 = { by axiom 12 (codomain1) }
% 294.22/37.65 addition(multiplication(X, Y), zero)
% 294.22/37.65 = { by axiom 6 (additive_identity) }
% 294.22/37.65 multiplication(X, Y)
% 294.22/37.65
% 294.22/37.65 Lemma 39: multiplication(X, codomain(X)) = X.
% 294.22/37.65 Proof:
% 294.22/37.65 multiplication(X, codomain(X))
% 294.22/37.65 = { by lemma 38 R->L }
% 294.22/37.65 multiplication(X, addition(codomain(X), coantidomain(X)))
% 294.22/37.65 = { by lemma 33 }
% 294.22/37.65 multiplication(X, one)
% 294.22/37.65 = { by axiom 8 (multiplicative_right_identity) }
% 294.22/37.65 X
% 294.22/37.65
% 294.22/37.65 Lemma 40: codomain(coantidomain(X)) = coantidomain(X).
% 294.22/37.65 Proof:
% 294.22/37.65 codomain(coantidomain(X))
% 294.22/37.65 = { by lemma 36 R->L }
% 294.22/37.65 coantidomain(codomain(X))
% 294.22/37.65 = { by axiom 10 (multiplicative_left_identity) R->L }
% 294.22/37.65 multiplication(one, coantidomain(codomain(X)))
% 294.22/37.65 = { by lemma 33 R->L }
% 294.22/37.65 multiplication(addition(codomain(X), coantidomain(X)), coantidomain(codomain(X)))
% 294.22/37.65 = { by lemma 37 }
% 294.22/37.65 multiplication(coantidomain(X), coantidomain(codomain(X)))
% 294.22/37.65 = { by lemma 36 }
% 294.22/37.65 multiplication(coantidomain(X), codomain(coantidomain(X)))
% 294.22/37.65 = { by lemma 39 }
% 294.22/37.65 coantidomain(X)
% 294.22/37.65
% 294.22/37.65 Lemma 41: antidomain(antidomain(multiplication(X, antidomain(antidomain(Y))))) = forward_diamond(X, Y).
% 294.22/37.65 Proof:
% 294.22/37.65 antidomain(antidomain(multiplication(X, antidomain(antidomain(Y)))))
% 294.22/37.65 = { by axiom 1 (domain4) R->L }
% 294.22/37.65 antidomain(antidomain(multiplication(X, domain(Y))))
% 294.22/37.65 = { by axiom 1 (domain4) R->L }
% 294.22/37.65 domain(multiplication(X, domain(Y)))
% 294.22/37.65 = { by axiom 18 (forward_diamond) R->L }
% 294.22/37.65 forward_diamond(X, Y)
% 294.22/37.65
% 294.22/37.65 Lemma 42: forward_diamond(X, one) = antidomain(antidomain(X)).
% 294.22/37.65 Proof:
% 294.22/37.65 forward_diamond(X, one)
% 294.22/37.65 = { by lemma 41 R->L }
% 294.22/37.65 antidomain(antidomain(multiplication(X, antidomain(antidomain(one)))))
% 294.22/37.65 = { by lemma 28 }
% 294.22/37.65 antidomain(antidomain(multiplication(X, antidomain(zero))))
% 294.22/37.65 = { by lemma 31 }
% 294.22/37.65 antidomain(antidomain(multiplication(X, one)))
% 294.22/37.65 = { by axiom 8 (multiplicative_right_identity) }
% 294.22/37.65 antidomain(antidomain(X))
% 294.22/37.65
% 294.22/37.65 Lemma 43: backward_diamond(X, one) = codomain(X).
% 294.22/37.65 Proof:
% 294.22/37.65 backward_diamond(X, one)
% 294.22/37.65 = { by axiom 16 (backward_diamond) }
% 294.22/37.65 codomain(multiplication(codomain(one), X))
% 294.22/37.65 = { by lemma 34 }
% 294.22/37.65 codomain(multiplication(one, X))
% 294.22/37.65 = { by axiom 10 (multiplicative_left_identity) }
% 294.22/37.65 codomain(X)
% 294.22/37.65
% 294.22/37.65 Lemma 44: addition(X, addition(X, Y)) = addition(X, Y).
% 294.22/37.65 Proof:
% 294.22/37.65 addition(X, addition(X, Y))
% 294.22/37.65 = { by axiom 19 (additive_associativity) }
% 294.22/37.65 addition(addition(X, X), Y)
% 294.22/37.65 = { by axiom 4 (additive_idempotence) }
% 294.22/37.65 addition(X, Y)
% 294.22/37.65
% 294.22/37.65 Lemma 45: addition(one, coantidomain(X)) = one.
% 294.22/37.65 Proof:
% 294.22/37.65 addition(one, coantidomain(X))
% 294.22/37.65 = { by axiom 5 (additive_commutativity) R->L }
% 294.22/37.65 addition(coantidomain(X), one)
% 294.22/37.65 = { by axiom 14 (ifeq_axiom) R->L }
% 294.22/37.65 ifeq(true, true, addition(coantidomain(X), one), one)
% 294.22/37.65 = { by axiom 26 (order) R->L }
% 294.22/37.65 ifeq(ifeq2(addition(coantidomain(X), addition(coantidomain(X), codomain(X))), addition(coantidomain(X), codomain(X)), leq(coantidomain(X), addition(coantidomain(X), codomain(X))), true), true, addition(coantidomain(X), one), one)
% 294.22/37.65 = { by lemma 44 }
% 294.22/37.65 ifeq(ifeq2(addition(coantidomain(X), codomain(X)), addition(coantidomain(X), codomain(X)), leq(coantidomain(X), addition(coantidomain(X), codomain(X))), true), true, addition(coantidomain(X), one), one)
% 294.22/37.65 = { by axiom 15 (ifeq_axiom) }
% 294.22/37.65 ifeq(leq(coantidomain(X), addition(coantidomain(X), codomain(X))), true, addition(coantidomain(X), one), one)
% 294.22/37.65 = { by axiom 5 (additive_commutativity) }
% 294.22/37.65 ifeq(leq(coantidomain(X), addition(codomain(X), coantidomain(X))), true, addition(coantidomain(X), one), one)
% 294.22/37.65 = { by lemma 33 }
% 294.22/37.65 ifeq(leq(coantidomain(X), one), true, addition(coantidomain(X), one), one)
% 294.22/37.65 = { by axiom 25 (order_1) }
% 294.22/37.65 one
% 294.22/37.65
% 294.22/37.65 Lemma 46: addition(domain(X), antidomain(domain(X))) = one.
% 294.22/37.65 Proof:
% 294.22/37.65 addition(domain(X), antidomain(domain(X)))
% 294.22/37.65 = { by axiom 1 (domain4) }
% 294.22/37.65 addition(domain(X), antidomain(antidomain(antidomain(X))))
% 294.22/37.65 = { by axiom 1 (domain4) }
% 294.22/37.65 addition(antidomain(antidomain(X)), antidomain(antidomain(antidomain(X))))
% 294.22/37.65 = { by lemma 30 }
% 294.22/37.65 one
% 294.22/37.65
% 294.22/37.65 Lemma 47: addition(domain(X), one) = one.
% 294.22/37.65 Proof:
% 294.22/37.65 addition(domain(X), one)
% 294.22/37.65 = { by lemma 46 R->L }
% 294.22/37.65 addition(domain(X), addition(domain(X), antidomain(domain(X))))
% 294.22/37.65 = { by lemma 44 }
% 294.22/37.65 addition(domain(X), antidomain(domain(X)))
% 294.22/37.65 = { by lemma 46 }
% 294.22/37.65 one
% 294.22/37.65
% 294.22/37.65 Lemma 48: multiplication(antidomain(X), addition(X, Y)) = multiplication(antidomain(X), Y).
% 294.22/37.65 Proof:
% 294.22/37.65 multiplication(antidomain(X), addition(X, Y))
% 294.22/37.65 = { by axiom 5 (additive_commutativity) R->L }
% 294.22/37.65 multiplication(antidomain(X), addition(Y, X))
% 294.22/37.65 = { by axiom 22 (right_distributivity) }
% 294.22/37.65 addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X))
% 294.22/37.65 = { by axiom 11 (domain1) }
% 294.22/37.65 addition(multiplication(antidomain(X), Y), zero)
% 294.22/37.65 = { by axiom 6 (additive_identity) }
% 294.22/37.65 multiplication(antidomain(X), Y)
% 294.22/37.65
% 294.22/37.65 Lemma 49: multiplication(antidomain(X), addition(Y, X)) = multiplication(antidomain(X), Y).
% 294.22/37.65 Proof:
% 294.22/37.65 multiplication(antidomain(X), addition(Y, X))
% 294.22/37.65 = { by axiom 5 (additive_commutativity) R->L }
% 294.22/37.65 multiplication(antidomain(X), addition(X, Y))
% 294.22/37.65 = { by lemma 48 }
% 294.22/37.65 multiplication(antidomain(X), Y)
% 294.22/37.65
% 294.22/37.66 Lemma 50: multiplication(addition(X, antidomain(Y)), Y) = multiplication(X, Y).
% 294.22/37.66 Proof:
% 294.22/37.66 multiplication(addition(X, antidomain(Y)), Y)
% 294.22/37.66 = { by axiom 23 (left_distributivity) }
% 294.22/37.66 addition(multiplication(X, Y), multiplication(antidomain(Y), Y))
% 294.22/37.66 = { by axiom 11 (domain1) }
% 294.22/37.66 addition(multiplication(X, Y), zero)
% 294.22/37.66 = { by axiom 6 (additive_identity) }
% 294.22/37.66 multiplication(X, Y)
% 294.22/37.66
% 294.22/37.66 Lemma 51: multiplication(addition(antidomain(X), Y), X) = multiplication(Y, X).
% 294.22/37.66 Proof:
% 294.22/37.66 multiplication(addition(antidomain(X), Y), X)
% 294.22/37.66 = { by axiom 5 (additive_commutativity) R->L }
% 294.22/37.66 multiplication(addition(Y, antidomain(X)), X)
% 294.22/37.66 = { by lemma 50 }
% 294.22/37.66 multiplication(Y, X)
% 294.22/37.66
% 294.22/37.66 Lemma 52: multiplication(antidomain(antidomain(X)), X) = X.
% 294.22/37.66 Proof:
% 294.22/37.66 multiplication(antidomain(antidomain(X)), X)
% 294.22/37.66 = { by lemma 51 R->L }
% 294.22/37.66 multiplication(addition(antidomain(X), antidomain(antidomain(X))), X)
% 294.22/37.66 = { by lemma 30 }
% 294.22/37.66 multiplication(one, X)
% 294.22/37.66 = { by axiom 10 (multiplicative_left_identity) }
% 294.22/37.66 X
% 294.22/37.66
% 294.22/37.66 Lemma 53: antidomain(antidomain(antidomain(X))) = antidomain(X).
% 294.22/37.66 Proof:
% 294.22/37.66 antidomain(antidomain(antidomain(X)))
% 294.22/37.66 = { by axiom 8 (multiplicative_right_identity) R->L }
% 294.22/37.66 multiplication(antidomain(antidomain(antidomain(X))), one)
% 294.22/37.66 = { by lemma 30 R->L }
% 294.22/37.66 multiplication(antidomain(antidomain(antidomain(X))), addition(antidomain(X), antidomain(antidomain(X))))
% 294.22/37.66 = { by lemma 49 }
% 294.22/37.66 multiplication(antidomain(antidomain(antidomain(X))), antidomain(X))
% 294.22/37.66 = { by lemma 52 }
% 294.22/37.66 antidomain(X)
% 294.22/37.66
% 294.22/37.66 Lemma 54: forward_diamond(X, antidomain(antidomain(Y))) = forward_diamond(X, Y).
% 294.22/37.66 Proof:
% 294.22/37.66 forward_diamond(X, antidomain(antidomain(Y)))
% 294.22/37.66 = { by lemma 41 R->L }
% 294.22/37.66 antidomain(antidomain(multiplication(X, antidomain(antidomain(antidomain(antidomain(Y)))))))
% 294.22/37.66 = { by lemma 53 }
% 294.22/37.66 antidomain(antidomain(multiplication(X, antidomain(antidomain(Y)))))
% 294.22/37.66 = { by lemma 41 }
% 294.22/37.66 forward_diamond(X, Y)
% 294.22/37.66
% 294.22/37.66 Lemma 55: antidomain(antidomain(antidomain(forward_diamond(X, antidomain(antidomain(antidomain(Y))))))) = forward_box(X, Y).
% 294.22/37.66 Proof:
% 294.22/37.66 antidomain(antidomain(antidomain(forward_diamond(X, antidomain(antidomain(antidomain(Y)))))))
% 294.22/37.66 = { by lemma 35 R->L }
% 294.22/37.66 antidomain(antidomain(antidomain(forward_diamond(X, c(Y)))))
% 294.22/37.66 = { by lemma 35 R->L }
% 294.22/37.66 c(forward_diamond(X, c(Y)))
% 294.22/37.66 = { by axiom 17 (forward_box) R->L }
% 294.22/37.66 forward_box(X, Y)
% 294.22/37.66
% 294.22/37.66 Lemma 56: antidomain(forward_diamond(X, antidomain(Y))) = forward_box(X, Y).
% 294.22/37.66 Proof:
% 294.22/37.66 antidomain(forward_diamond(X, antidomain(Y)))
% 294.22/37.66 = { by lemma 53 R->L }
% 294.22/37.66 antidomain(forward_diamond(X, antidomain(antidomain(antidomain(Y)))))
% 294.22/37.66 = { by lemma 53 R->L }
% 294.22/37.66 antidomain(antidomain(antidomain(forward_diamond(X, antidomain(antidomain(antidomain(Y)))))))
% 294.22/37.66 = { by lemma 55 }
% 294.22/37.66 forward_box(X, Y)
% 294.22/37.66
% 294.22/37.66 Lemma 57: antidomain(forward_diamond(X, Y)) = forward_box(X, antidomain(Y)).
% 294.22/37.66 Proof:
% 294.22/37.66 antidomain(forward_diamond(X, Y))
% 294.22/37.66 = { by lemma 54 R->L }
% 294.22/37.66 antidomain(forward_diamond(X, antidomain(antidomain(Y))))
% 294.22/37.66 = { by lemma 56 }
% 294.22/37.66 forward_box(X, antidomain(Y))
% 294.22/37.66
% 294.22/37.66 Lemma 58: codomain(backward_diamond(X, Y)) = backward_diamond(X, Y).
% 294.22/37.66 Proof:
% 294.22/37.66 codomain(backward_diamond(X, Y))
% 294.22/37.66 = { by axiom 16 (backward_diamond) }
% 294.22/37.66 codomain(codomain(multiplication(codomain(Y), X)))
% 294.22/37.66 = { by axiom 3 (codomain4) }
% 294.22/37.66 codomain(coantidomain(coantidomain(multiplication(codomain(Y), X))))
% 294.22/37.66 = { by lemma 40 }
% 294.22/37.66 coantidomain(coantidomain(multiplication(codomain(Y), X)))
% 294.22/37.66 = { by axiom 3 (codomain4) R->L }
% 294.22/37.66 codomain(multiplication(codomain(Y), X))
% 294.22/37.66 = { by axiom 16 (backward_diamond) R->L }
% 294.22/37.66 backward_diamond(X, Y)
% 294.22/37.66
% 294.22/37.66 Lemma 59: multiplication(antidomain(antidomain(X)), antidomain(Y)) = domain_difference(X, Y).
% 294.22/37.66 Proof:
% 294.22/37.66 multiplication(antidomain(antidomain(X)), antidomain(Y))
% 294.22/37.66 = { by axiom 1 (domain4) R->L }
% 294.22/37.66 multiplication(domain(X), antidomain(Y))
% 294.22/37.66 = { by axiom 13 (domain_difference) R->L }
% 294.22/37.66 domain_difference(X, Y)
% 294.22/37.66
% 294.22/37.66 Lemma 60: domain_difference(domain(X), Y) = domain_difference(X, Y).
% 294.22/37.66 Proof:
% 294.22/37.66 domain_difference(domain(X), Y)
% 294.22/37.66 = { by lemma 59 R->L }
% 294.22/37.66 multiplication(antidomain(antidomain(domain(X))), antidomain(Y))
% 294.22/37.66 = { by axiom 8 (multiplicative_right_identity) R->L }
% 294.22/37.66 multiplication(antidomain(multiplication(antidomain(domain(X)), one)), antidomain(Y))
% 294.22/37.66 = { by lemma 30 R->L }
% 294.22/37.66 multiplication(antidomain(multiplication(antidomain(domain(X)), addition(antidomain(X), antidomain(antidomain(X))))), antidomain(Y))
% 294.22/37.66 = { by axiom 1 (domain4) R->L }
% 294.22/37.66 multiplication(antidomain(multiplication(antidomain(domain(X)), addition(antidomain(X), domain(X)))), antidomain(Y))
% 294.22/37.66 = { by axiom 5 (additive_commutativity) }
% 294.22/37.66 multiplication(antidomain(multiplication(antidomain(domain(X)), addition(domain(X), antidomain(X)))), antidomain(Y))
% 294.22/37.66 = { by lemma 48 }
% 294.22/37.66 multiplication(antidomain(multiplication(antidomain(domain(X)), antidomain(X))), antidomain(Y))
% 294.22/37.66 = { by axiom 1 (domain4) }
% 294.22/37.66 multiplication(antidomain(multiplication(antidomain(antidomain(antidomain(X))), antidomain(X))), antidomain(Y))
% 294.22/37.66 = { by lemma 52 }
% 294.22/37.66 multiplication(antidomain(antidomain(X)), antidomain(Y))
% 294.22/37.66 = { by lemma 59 }
% 294.22/37.66 domain_difference(X, Y)
% 294.22/37.66
% 294.22/37.66 Lemma 61: addition(Y, addition(X, Z)) = addition(X, addition(Y, Z)).
% 294.22/37.66 Proof:
% 294.22/37.66 addition(Y, addition(X, Z))
% 294.22/37.66 = { by axiom 5 (additive_commutativity) R->L }
% 294.22/37.66 addition(addition(X, Z), Y)
% 294.22/37.66 = { by axiom 19 (additive_associativity) R->L }
% 294.22/37.66 addition(X, addition(Z, Y))
% 294.22/37.66 = { by axiom 5 (additive_commutativity) }
% 294.22/37.66 addition(X, addition(Y, Z))
% 294.22/37.66
% 294.22/37.66 Lemma 62: addition(Z, addition(X, Y)) = addition(X, addition(Y, Z)).
% 294.22/37.66 Proof:
% 294.22/37.66 addition(Z, addition(X, Y))
% 294.22/37.66 = { by lemma 61 }
% 294.22/37.66 addition(X, addition(Z, Y))
% 294.22/37.66 = { by axiom 5 (additive_commutativity) }
% 294.22/37.66 addition(X, addition(Y, Z))
% 294.22/37.66
% 294.22/37.66 Lemma 63: addition(forward_box(X, Y), antidomain(forward_box(X, Y))) = one.
% 294.22/37.66 Proof:
% 294.22/37.66 addition(forward_box(X, Y), antidomain(forward_box(X, Y)))
% 294.22/37.66 = { by lemma 55 R->L }
% 294.22/37.66 addition(forward_box(X, Y), antidomain(antidomain(antidomain(antidomain(forward_diamond(X, antidomain(antidomain(antidomain(Y)))))))))
% 294.22/37.66 = { by lemma 55 R->L }
% 294.22/37.66 addition(antidomain(antidomain(antidomain(forward_diamond(X, antidomain(antidomain(antidomain(Y))))))), antidomain(antidomain(antidomain(antidomain(forward_diamond(X, antidomain(antidomain(antidomain(Y)))))))))
% 294.22/37.66 = { by lemma 30 }
% 294.22/37.66 one
% 294.22/37.66
% 294.22/37.66 Lemma 64: addition(forward_box(X, Y), one) = one.
% 294.22/37.66 Proof:
% 294.22/37.66 addition(forward_box(X, Y), one)
% 294.22/37.66 = { by lemma 63 R->L }
% 294.22/37.66 addition(forward_box(X, Y), addition(forward_box(X, Y), antidomain(forward_box(X, Y))))
% 294.22/37.66 = { by lemma 44 }
% 294.22/37.66 addition(forward_box(X, Y), antidomain(forward_box(X, Y)))
% 294.22/37.66 = { by lemma 63 }
% 294.22/37.66 one
% 294.22/37.66
% 294.22/37.66 Lemma 65: multiplication(addition(X, one), Y) = addition(Y, multiplication(X, Y)).
% 294.22/37.66 Proof:
% 294.22/37.66 multiplication(addition(X, one), Y)
% 294.22/37.66 = { by axiom 5 (additive_commutativity) R->L }
% 294.22/37.66 multiplication(addition(one, X), Y)
% 294.22/37.66 = { by axiom 23 (left_distributivity) }
% 294.22/37.66 addition(multiplication(one, Y), multiplication(X, Y))
% 294.22/37.66 = { by axiom 10 (multiplicative_left_identity) }
% 294.22/37.66 addition(Y, multiplication(X, Y))
% 294.22/37.66
% 294.22/37.66 Lemma 66: antidomain(multiplication(X, antidomain(antidomain(Y)))) = forward_box(X, antidomain(Y)).
% 294.22/37.66 Proof:
% 294.22/37.66 antidomain(multiplication(X, antidomain(antidomain(Y))))
% 294.22/37.66 = { by lemma 53 R->L }
% 294.22/37.66 antidomain(antidomain(antidomain(multiplication(X, antidomain(antidomain(Y))))))
% 294.22/37.66 = { by lemma 41 }
% 294.22/37.66 antidomain(forward_diamond(X, Y))
% 294.22/37.66 = { by lemma 57 }
% 294.22/37.66 forward_box(X, antidomain(Y))
% 294.22/37.66
% 294.22/37.66 Lemma 67: forward_box(X, antidomain(antidomain(Y))) = forward_box(X, Y).
% 294.22/37.66 Proof:
% 294.22/37.66 forward_box(X, antidomain(antidomain(Y)))
% 294.22/37.66 = { by lemma 56 R->L }
% 294.22/37.66 antidomain(forward_diamond(X, antidomain(antidomain(antidomain(Y)))))
% 294.22/37.66 = { by lemma 53 R->L }
% 294.22/37.66 antidomain(antidomain(antidomain(forward_diamond(X, antidomain(antidomain(antidomain(Y)))))))
% 294.22/37.66 = { by lemma 55 }
% 294.22/37.66 forward_box(X, Y)
% 294.22/37.66
% 294.22/37.66 Lemma 68: antidomain(multiplication(X, antidomain(Y))) = forward_box(X, Y).
% 294.22/37.66 Proof:
% 294.22/37.66 antidomain(multiplication(X, antidomain(Y)))
% 294.22/37.66 = { by lemma 53 R->L }
% 294.22/37.66 antidomain(multiplication(X, antidomain(antidomain(antidomain(Y)))))
% 294.22/37.66 = { by lemma 66 }
% 294.22/37.66 forward_box(X, antidomain(antidomain(Y)))
% 294.22/37.66 = { by lemma 67 }
% 294.22/37.66 forward_box(X, Y)
% 294.22/37.66
% 294.22/37.66 Lemma 69: antidomain(antidomain(forward_diamond(X, Y))) = forward_diamond(X, Y).
% 294.22/37.66 Proof:
% 294.22/37.66 antidomain(antidomain(forward_diamond(X, Y)))
% 294.22/37.66 = { by axiom 8 (multiplicative_right_identity) R->L }
% 294.22/37.66 multiplication(antidomain(antidomain(forward_diamond(X, Y))), one)
% 294.22/37.66 = { by lemma 30 R->L }
% 294.22/37.66 multiplication(antidomain(antidomain(forward_diamond(X, Y))), addition(antidomain(antidomain(multiplication(X, antidomain(antidomain(Y))))), antidomain(antidomain(antidomain(multiplication(X, antidomain(antidomain(Y))))))))
% 294.22/37.66 = { by lemma 41 }
% 294.22/37.66 multiplication(antidomain(antidomain(forward_diamond(X, Y))), addition(forward_diamond(X, Y), antidomain(antidomain(antidomain(multiplication(X, antidomain(antidomain(Y))))))))
% 294.22/37.66 = { by lemma 41 }
% 294.22/37.66 multiplication(antidomain(antidomain(forward_diamond(X, Y))), addition(forward_diamond(X, Y), antidomain(forward_diamond(X, Y))))
% 294.22/37.66 = { by lemma 49 }
% 294.22/37.66 multiplication(antidomain(antidomain(forward_diamond(X, Y))), forward_diamond(X, Y))
% 294.22/37.66 = { by lemma 52 }
% 294.22/37.66 forward_diamond(X, Y)
% 294.22/37.66
% 294.22/37.66 Lemma 70: antidomain(antidomain(forward_box(X, Y))) = forward_box(X, Y).
% 294.22/37.66 Proof:
% 294.22/37.66 antidomain(antidomain(forward_box(X, Y)))
% 294.22/37.66 = { by axiom 8 (multiplicative_right_identity) R->L }
% 294.22/37.66 multiplication(antidomain(antidomain(forward_box(X, Y))), one)
% 294.22/37.66 = { by lemma 63 R->L }
% 294.22/37.66 multiplication(antidomain(antidomain(forward_box(X, Y))), addition(forward_box(X, Y), antidomain(forward_box(X, Y))))
% 294.22/37.66 = { by lemma 49 }
% 294.22/37.66 multiplication(antidomain(antidomain(forward_box(X, Y))), forward_box(X, Y))
% 294.22/37.66 = { by lemma 52 }
% 294.22/37.66 forward_box(X, Y)
% 294.22/37.66
% 294.22/37.66 Lemma 71: antidomain(domain_difference(domain(X), Y)) = forward_box(domain(X), Y).
% 294.22/37.66 Proof:
% 294.22/37.66 antidomain(domain_difference(domain(X), Y))
% 294.22/37.66 = { by lemma 60 }
% 294.22/37.66 antidomain(domain_difference(X, Y))
% 294.22/37.66 = { by lemma 59 R->L }
% 294.22/37.66 antidomain(multiplication(antidomain(antidomain(X)), antidomain(Y)))
% 294.22/37.66 = { by axiom 1 (domain4) R->L }
% 294.22/37.66 antidomain(multiplication(domain(X), antidomain(Y)))
% 294.22/37.66 = { by lemma 68 }
% 294.22/37.66 forward_box(domain(X), Y)
% 294.22/37.66
% 294.22/37.66 Lemma 72: coantidomain(multiplication(codomain(X), Y)) = coantidomain(backward_diamond(Y, X)).
% 294.22/37.66 Proof:
% 294.22/37.66 coantidomain(multiplication(codomain(X), Y))
% 294.22/37.66 = { by lemma 40 R->L }
% 294.22/37.66 codomain(coantidomain(multiplication(codomain(X), Y)))
% 294.22/37.66 = { by lemma 36 R->L }
% 294.22/37.66 coantidomain(codomain(multiplication(codomain(X), Y)))
% 294.22/37.66 = { by axiom 16 (backward_diamond) R->L }
% 294.22/37.67 coantidomain(backward_diamond(Y, X))
% 294.22/37.67
% 294.22/37.67 Lemma 73: multiplication(codomain(antidomain(X)), addition(X, Y)) = multiplication(codomain(antidomain(X)), Y).
% 294.22/37.67 Proof:
% 294.22/37.67 multiplication(codomain(antidomain(X)), addition(X, Y))
% 294.22/37.67 = { by axiom 22 (right_distributivity) }
% 294.22/37.67 addition(multiplication(codomain(antidomain(X)), X), multiplication(codomain(antidomain(X)), Y))
% 294.22/37.67 = { by lemma 39 R->L }
% 294.22/37.67 addition(multiplication(multiplication(codomain(antidomain(X)), X), codomain(multiplication(codomain(antidomain(X)), X))), multiplication(codomain(antidomain(X)), Y))
% 294.22/37.67 = { by axiom 16 (backward_diamond) R->L }
% 294.22/37.67 addition(multiplication(multiplication(codomain(antidomain(X)), X), backward_diamond(X, antidomain(X))), multiplication(codomain(antidomain(X)), Y))
% 294.22/37.67 = { by axiom 20 (multiplicative_associativity) R->L }
% 294.22/37.67 addition(multiplication(codomain(antidomain(X)), multiplication(X, backward_diamond(X, antidomain(X)))), multiplication(codomain(antidomain(X)), Y))
% 294.22/37.67 = { by lemma 58 R->L }
% 294.22/37.67 addition(multiplication(codomain(antidomain(X)), multiplication(X, codomain(backward_diamond(X, antidomain(X))))), multiplication(codomain(antidomain(X)), Y))
% 294.22/37.67 = { by axiom 3 (codomain4) }
% 294.22/37.67 addition(multiplication(codomain(antidomain(X)), multiplication(X, coantidomain(coantidomain(backward_diamond(X, antidomain(X)))))), multiplication(codomain(antidomain(X)), Y))
% 294.22/37.67 = { by lemma 72 R->L }
% 294.22/37.67 addition(multiplication(codomain(antidomain(X)), multiplication(X, coantidomain(coantidomain(multiplication(codomain(antidomain(X)), X))))), multiplication(codomain(antidomain(X)), Y))
% 294.22/37.67 = { by axiom 3 (codomain4) }
% 294.22/37.67 addition(multiplication(codomain(antidomain(X)), multiplication(X, coantidomain(coantidomain(multiplication(coantidomain(coantidomain(antidomain(X))), X))))), multiplication(codomain(antidomain(X)), Y))
% 294.22/37.67 = { by axiom 27 (codomain2) R->L }
% 294.22/37.67 addition(multiplication(codomain(antidomain(X)), multiplication(X, coantidomain(addition(coantidomain(multiplication(antidomain(X), X)), coantidomain(multiplication(coantidomain(coantidomain(antidomain(X))), X)))))), multiplication(codomain(antidomain(X)), Y))
% 294.22/37.67 = { by axiom 3 (codomain4) R->L }
% 294.22/37.67 addition(multiplication(codomain(antidomain(X)), multiplication(X, coantidomain(addition(coantidomain(multiplication(antidomain(X), X)), coantidomain(multiplication(codomain(antidomain(X)), X)))))), multiplication(codomain(antidomain(X)), Y))
% 294.22/37.67 = { by lemma 72 }
% 294.22/37.67 addition(multiplication(codomain(antidomain(X)), multiplication(X, coantidomain(addition(coantidomain(multiplication(antidomain(X), X)), coantidomain(backward_diamond(X, antidomain(X))))))), multiplication(codomain(antidomain(X)), Y))
% 294.22/37.67 = { by axiom 5 (additive_commutativity) }
% 294.22/37.67 addition(multiplication(codomain(antidomain(X)), multiplication(X, coantidomain(addition(coantidomain(backward_diamond(X, antidomain(X))), coantidomain(multiplication(antidomain(X), X)))))), multiplication(codomain(antidomain(X)), Y))
% 294.22/37.67 = { by axiom 11 (domain1) }
% 294.22/37.67 addition(multiplication(codomain(antidomain(X)), multiplication(X, coantidomain(addition(coantidomain(backward_diamond(X, antidomain(X))), coantidomain(zero))))), multiplication(codomain(antidomain(X)), Y))
% 294.22/37.67 = { by lemma 32 R->L }
% 294.22/37.67 addition(multiplication(codomain(antidomain(X)), multiplication(X, coantidomain(addition(coantidomain(backward_diamond(X, antidomain(X))), coantidomain(coantidomain(one)))))), multiplication(codomain(antidomain(X)), Y))
% 294.22/37.67 = { by axiom 3 (codomain4) R->L }
% 294.22/37.67 addition(multiplication(codomain(antidomain(X)), multiplication(X, coantidomain(addition(coantidomain(backward_diamond(X, antidomain(X))), codomain(one))))), multiplication(codomain(antidomain(X)), Y))
% 294.22/37.67 = { by lemma 34 }
% 294.22/37.67 addition(multiplication(codomain(antidomain(X)), multiplication(X, coantidomain(addition(coantidomain(backward_diamond(X, antidomain(X))), one)))), multiplication(codomain(antidomain(X)), Y))
% 294.22/37.67 = { by axiom 5 (additive_commutativity) }
% 294.22/37.67 addition(multiplication(codomain(antidomain(X)), multiplication(X, coantidomain(addition(one, coantidomain(backward_diamond(X, antidomain(X))))))), multiplication(codomain(antidomain(X)), Y))
% 294.22/37.67 = { by lemma 45 }
% 294.22/37.67 addition(multiplication(codomain(antidomain(X)), multiplication(X, coantidomain(one))), multiplication(codomain(antidomain(X)), Y))
% 294.22/37.67 = { by lemma 32 }
% 294.22/37.67 addition(multiplication(codomain(antidomain(X)), multiplication(X, zero)), multiplication(codomain(antidomain(X)), Y))
% 294.22/37.67 = { by axiom 9 (right_annihilation) }
% 294.22/37.67 addition(multiplication(codomain(antidomain(X)), zero), multiplication(codomain(antidomain(X)), Y))
% 294.22/37.67 = { by axiom 9 (right_annihilation) }
% 294.22/37.67 addition(zero, multiplication(codomain(antidomain(X)), Y))
% 294.22/37.67 = { by lemma 29 }
% 294.22/37.67 multiplication(codomain(antidomain(X)), Y)
% 294.22/37.67
% 294.22/37.67 Lemma 74: codomain(antidomain(forward_box(X, Y))) = antidomain(forward_box(X, Y)).
% 294.22/37.67 Proof:
% 294.22/37.67 codomain(antidomain(forward_box(X, Y)))
% 294.22/37.67 = { by axiom 8 (multiplicative_right_identity) R->L }
% 294.22/37.67 multiplication(codomain(antidomain(forward_box(X, Y))), one)
% 294.22/37.67 = { by lemma 63 R->L }
% 294.22/37.67 multiplication(codomain(antidomain(forward_box(X, Y))), addition(forward_box(X, Y), antidomain(forward_box(X, Y))))
% 294.22/37.67 = { by lemma 73 }
% 294.22/37.67 multiplication(codomain(antidomain(forward_box(X, Y))), antidomain(forward_box(X, Y)))
% 294.22/37.67 = { by lemma 51 R->L }
% 294.22/37.67 multiplication(addition(antidomain(antidomain(forward_box(X, Y))), codomain(antidomain(forward_box(X, Y)))), antidomain(forward_box(X, Y)))
% 294.22/37.67 = { by lemma 70 }
% 294.22/37.67 multiplication(addition(forward_box(X, Y), codomain(antidomain(forward_box(X, Y)))), antidomain(forward_box(X, Y)))
% 294.22/37.67 = { by axiom 5 (additive_commutativity) R->L }
% 294.22/37.67 multiplication(addition(codomain(antidomain(forward_box(X, Y))), forward_box(X, Y)), antidomain(forward_box(X, Y)))
% 294.22/37.67 = { by axiom 8 (multiplicative_right_identity) R->L }
% 294.22/37.67 multiplication(addition(codomain(antidomain(forward_box(X, Y))), multiplication(forward_box(X, Y), one)), antidomain(forward_box(X, Y)))
% 294.22/37.67 = { by lemma 45 R->L }
% 294.22/37.67 multiplication(addition(codomain(antidomain(forward_box(X, Y))), multiplication(forward_box(X, Y), addition(one, coantidomain(antidomain(forward_box(X, Y)))))), antidomain(forward_box(X, Y)))
% 294.22/37.67 = { by axiom 22 (right_distributivity) }
% 294.22/37.67 multiplication(addition(codomain(antidomain(forward_box(X, Y))), addition(multiplication(forward_box(X, Y), one), multiplication(forward_box(X, Y), coantidomain(antidomain(forward_box(X, Y)))))), antidomain(forward_box(X, Y)))
% 294.22/37.67 = { by axiom 8 (multiplicative_right_identity) }
% 294.22/37.67 multiplication(addition(codomain(antidomain(forward_box(X, Y))), addition(forward_box(X, Y), multiplication(forward_box(X, Y), coantidomain(antidomain(forward_box(X, Y)))))), antidomain(forward_box(X, Y)))
% 294.22/37.67 = { by lemma 37 R->L }
% 294.22/37.67 multiplication(addition(codomain(antidomain(forward_box(X, Y))), addition(forward_box(X, Y), multiplication(addition(antidomain(forward_box(X, Y)), forward_box(X, Y)), coantidomain(antidomain(forward_box(X, Y)))))), antidomain(forward_box(X, Y)))
% 294.22/37.67 = { by axiom 5 (additive_commutativity) }
% 294.22/37.67 multiplication(addition(codomain(antidomain(forward_box(X, Y))), addition(forward_box(X, Y), multiplication(addition(forward_box(X, Y), antidomain(forward_box(X, Y))), coantidomain(antidomain(forward_box(X, Y)))))), antidomain(forward_box(X, Y)))
% 294.22/37.67 = { by lemma 63 }
% 294.22/37.67 multiplication(addition(codomain(antidomain(forward_box(X, Y))), addition(forward_box(X, Y), multiplication(one, coantidomain(antidomain(forward_box(X, Y)))))), antidomain(forward_box(X, Y)))
% 294.22/37.67 = { by axiom 10 (multiplicative_left_identity) }
% 294.22/37.67 multiplication(addition(codomain(antidomain(forward_box(X, Y))), addition(forward_box(X, Y), coantidomain(antidomain(forward_box(X, Y))))), antidomain(forward_box(X, Y)))
% 294.22/37.67 = { by axiom 5 (additive_commutativity) R->L }
% 294.22/37.67 multiplication(addition(codomain(antidomain(forward_box(X, Y))), addition(coantidomain(antidomain(forward_box(X, Y))), forward_box(X, Y))), antidomain(forward_box(X, Y)))
% 294.22/37.67 = { by axiom 19 (additive_associativity) }
% 294.22/37.67 multiplication(addition(addition(codomain(antidomain(forward_box(X, Y))), coantidomain(antidomain(forward_box(X, Y)))), forward_box(X, Y)), antidomain(forward_box(X, Y)))
% 294.22/37.67 = { by lemma 33 }
% 294.22/37.67 multiplication(addition(one, forward_box(X, Y)), antidomain(forward_box(X, Y)))
% 294.22/37.67 = { by axiom 5 (additive_commutativity) }
% 294.22/37.67 multiplication(addition(forward_box(X, Y), one), antidomain(forward_box(X, Y)))
% 294.22/37.67 = { by lemma 64 }
% 294.22/37.67 multiplication(one, antidomain(forward_box(X, Y)))
% 294.22/37.67 = { by axiom 10 (multiplicative_left_identity) }
% 294.22/37.67 antidomain(forward_box(X, Y))
% 294.22/37.67
% 294.22/37.67 Lemma 75: multiplication(antidomain(codomain(X)), coantidomain(X)) = antidomain(codomain(X)).
% 294.22/37.67 Proof:
% 294.22/37.67 multiplication(antidomain(codomain(X)), coantidomain(X))
% 294.22/37.67 = { by lemma 48 R->L }
% 294.22/37.67 multiplication(antidomain(codomain(X)), addition(codomain(X), coantidomain(X)))
% 294.22/37.67 = { by lemma 33 }
% 294.22/37.67 multiplication(antidomain(codomain(X)), one)
% 294.22/37.67 = { by axiom 8 (multiplicative_right_identity) }
% 294.22/37.67 antidomain(codomain(X))
% 294.22/37.67
% 294.22/37.67 Lemma 76: coantidomain(antidomain(forward_box(X, Y))) = forward_box(X, Y).
% 294.22/37.67 Proof:
% 294.22/37.67 coantidomain(antidomain(forward_box(X, Y)))
% 294.22/37.67 = { by axiom 10 (multiplicative_left_identity) R->L }
% 294.22/37.67 multiplication(one, coantidomain(antidomain(forward_box(X, Y))))
% 294.22/37.67 = { by lemma 30 R->L }
% 294.22/37.67 multiplication(addition(antidomain(forward_box(X, Y)), antidomain(antidomain(forward_box(X, Y)))), coantidomain(antidomain(forward_box(X, Y))))
% 294.22/37.67 = { by lemma 37 }
% 294.22/37.67 multiplication(antidomain(antidomain(forward_box(X, Y))), coantidomain(antidomain(forward_box(X, Y))))
% 294.22/37.67 = { by lemma 74 R->L }
% 294.22/37.67 multiplication(antidomain(codomain(antidomain(forward_box(X, Y)))), coantidomain(antidomain(forward_box(X, Y))))
% 294.22/37.67 = { by lemma 75 }
% 294.22/37.67 antidomain(codomain(antidomain(forward_box(X, Y))))
% 294.22/37.67 = { by lemma 74 }
% 294.22/37.67 antidomain(antidomain(forward_box(X, Y)))
% 294.22/37.67 = { by lemma 70 }
% 294.22/37.67 forward_box(X, Y)
% 294.22/37.67
% 294.22/37.67 Lemma 77: multiplication(X, multiplication(codomain(X), Y)) = multiplication(X, Y).
% 294.22/37.67 Proof:
% 294.22/37.67 multiplication(X, multiplication(codomain(X), Y))
% 294.22/37.67 = { by axiom 20 (multiplicative_associativity) }
% 294.22/37.67 multiplication(multiplication(X, codomain(X)), Y)
% 294.22/37.67 = { by lemma 39 }
% 294.22/37.67 multiplication(X, Y)
% 294.22/37.67
% 294.22/37.67 Lemma 78: forward_box(X, forward_box(Y, backward_diamond(Y, X))) = one.
% 294.22/37.67 Proof:
% 294.22/37.67 forward_box(X, forward_box(Y, backward_diamond(Y, X)))
% 294.22/37.67 = { by lemma 68 R->L }
% 294.22/37.67 forward_box(X, antidomain(multiplication(Y, antidomain(backward_diamond(Y, X)))))
% 294.22/37.67 = { by lemma 57 R->L }
% 294.22/37.67 antidomain(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))))
% 294.22/37.67 = { by lemma 53 R->L }
% 294.22/37.67 antidomain(antidomain(antidomain(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))))))
% 294.22/37.67 = { by lemma 42 R->L }
% 294.22/37.67 antidomain(forward_diamond(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), one))
% 294.22/37.67 = { by lemma 31 R->L }
% 294.22/37.67 antidomain(forward_diamond(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), antidomain(zero)))
% 294.22/37.67 = { by lemma 56 }
% 294.22/37.67 forward_box(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), zero)
% 294.22/37.67 = { by axiom 9 (right_annihilation) R->L }
% 294.22/37.67 forward_box(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), multiplication(X, zero))
% 294.22/37.67 = { by axiom 9 (right_annihilation) R->L }
% 294.22/37.67 forward_box(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), multiplication(X, multiplication(Y, zero)))
% 294.22/37.67 = { by axiom 12 (codomain1) R->L }
% 294.22/37.67 forward_box(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), multiplication(X, multiplication(Y, multiplication(backward_diamond(Y, X), coantidomain(backward_diamond(Y, X))))))
% 294.22/37.67 = { by axiom 10 (multiplicative_left_identity) R->L }
% 294.22/37.67 forward_box(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), multiplication(X, multiplication(Y, multiplication(backward_diamond(Y, X), multiplication(one, coantidomain(backward_diamond(Y, X)))))))
% 294.22/37.68 = { by lemma 30 R->L }
% 294.22/37.68 forward_box(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), multiplication(X, multiplication(Y, multiplication(backward_diamond(Y, X), multiplication(addition(antidomain(codomain(backward_diamond(Y, X))), antidomain(antidomain(codomain(backward_diamond(Y, X))))), coantidomain(backward_diamond(Y, X)))))))
% 294.22/37.68 = { by lemma 44 R->L }
% 294.22/37.68 forward_box(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), multiplication(X, multiplication(Y, multiplication(backward_diamond(Y, X), multiplication(addition(antidomain(codomain(backward_diamond(Y, X))), addition(antidomain(codomain(backward_diamond(Y, X))), antidomain(antidomain(codomain(backward_diamond(Y, X)))))), coantidomain(backward_diamond(Y, X)))))))
% 294.22/37.68 = { by lemma 30 }
% 294.22/37.68 forward_box(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), multiplication(X, multiplication(Y, multiplication(backward_diamond(Y, X), multiplication(addition(antidomain(codomain(backward_diamond(Y, X))), one), coantidomain(backward_diamond(Y, X)))))))
% 294.22/37.68 = { by lemma 65 }
% 294.22/37.68 forward_box(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), multiplication(X, multiplication(Y, multiplication(backward_diamond(Y, X), addition(coantidomain(backward_diamond(Y, X)), multiplication(antidomain(codomain(backward_diamond(Y, X))), coantidomain(backward_diamond(Y, X))))))))
% 294.22/37.68 = { by lemma 75 }
% 294.22/37.68 forward_box(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), multiplication(X, multiplication(Y, multiplication(backward_diamond(Y, X), addition(coantidomain(backward_diamond(Y, X)), antidomain(codomain(backward_diamond(Y, X))))))))
% 294.22/37.68 = { by axiom 5 (additive_commutativity) R->L }
% 294.22/37.68 forward_box(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), multiplication(X, multiplication(Y, multiplication(backward_diamond(Y, X), addition(antidomain(codomain(backward_diamond(Y, X))), coantidomain(backward_diamond(Y, X)))))))
% 294.22/37.68 = { by lemma 38 }
% 294.22/37.68 forward_box(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), multiplication(X, multiplication(Y, multiplication(backward_diamond(Y, X), antidomain(codomain(backward_diamond(Y, X)))))))
% 294.22/37.68 = { by axiom 20 (multiplicative_associativity) }
% 294.22/37.68 forward_box(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), multiplication(X, multiplication(multiplication(Y, backward_diamond(Y, X)), antidomain(codomain(backward_diamond(Y, X))))))
% 294.22/37.68 = { by axiom 20 (multiplicative_associativity) }
% 294.22/37.68 forward_box(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), multiplication(multiplication(X, multiplication(Y, backward_diamond(Y, X))), antidomain(codomain(backward_diamond(Y, X)))))
% 294.22/37.68 = { by axiom 16 (backward_diamond) }
% 294.22/37.68 forward_box(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), multiplication(multiplication(X, multiplication(Y, codomain(multiplication(codomain(X), Y)))), antidomain(codomain(backward_diamond(Y, X)))))
% 294.22/37.68 = { by lemma 77 R->L }
% 294.22/37.68 forward_box(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), multiplication(multiplication(X, multiplication(codomain(X), multiplication(Y, codomain(multiplication(codomain(X), Y))))), antidomain(codomain(backward_diamond(Y, X)))))
% 294.22/37.68 = { by axiom 20 (multiplicative_associativity) }
% 294.22/37.68 forward_box(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), multiplication(multiplication(X, multiplication(multiplication(codomain(X), Y), codomain(multiplication(codomain(X), Y)))), antidomain(codomain(backward_diamond(Y, X)))))
% 294.22/37.68 = { by lemma 39 }
% 294.22/37.68 forward_box(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), multiplication(multiplication(X, multiplication(codomain(X), Y)), antidomain(codomain(backward_diamond(Y, X)))))
% 294.22/37.68 = { by lemma 77 }
% 294.22/37.68 forward_box(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), multiplication(multiplication(X, Y), antidomain(codomain(backward_diamond(Y, X)))))
% 294.22/37.68 = { by axiom 20 (multiplicative_associativity) R->L }
% 294.22/37.68 forward_box(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), multiplication(X, multiplication(Y, antidomain(codomain(backward_diamond(Y, X))))))
% 294.22/37.68 = { by lemma 58 }
% 294.22/37.68 forward_box(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X)))))
% 294.22/37.68 = { by lemma 69 R->L }
% 294.22/37.68 forward_box(antidomain(antidomain(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))))), multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X)))))
% 294.22/37.68 = { by lemma 67 R->L }
% 294.22/37.68 forward_box(antidomain(antidomain(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))))), antidomain(antidomain(multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X)))))))
% 294.22/37.68 = { by lemma 57 R->L }
% 294.22/37.68 antidomain(forward_diamond(antidomain(antidomain(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))))), antidomain(multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X)))))))
% 294.22/37.68 = { by lemma 41 R->L }
% 294.22/37.68 antidomain(antidomain(antidomain(multiplication(antidomain(antidomain(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))))), antidomain(antidomain(antidomain(multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X)))))))))))
% 294.22/37.68 = { by lemma 59 }
% 294.22/37.68 antidomain(antidomain(antidomain(domain_difference(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), antidomain(antidomain(multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X))))))))))
% 294.22/37.68 = { by lemma 53 }
% 294.22/37.68 antidomain(domain_difference(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), antidomain(antidomain(multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X))))))))
% 294.22/37.68 = { by lemma 59 R->L }
% 294.22/37.68 antidomain(multiplication(antidomain(antidomain(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))))), antidomain(antidomain(antidomain(multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X)))))))))
% 294.22/37.68 = { by lemma 53 }
% 294.22/37.68 antidomain(multiplication(antidomain(antidomain(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))))), antidomain(multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X)))))))
% 294.22/37.68 = { by lemma 59 }
% 294.22/37.68 antidomain(domain_difference(forward_diamond(X, multiplication(Y, antidomain(backward_diamond(Y, X)))), multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X))))))
% 294.22/37.68 = { by lemma 54 R->L }
% 294.22/37.68 antidomain(domain_difference(forward_diamond(X, antidomain(antidomain(multiplication(Y, antidomain(backward_diamond(Y, X)))))), multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X))))))
% 294.22/37.68 = { by lemma 69 R->L }
% 294.22/37.68 antidomain(domain_difference(antidomain(antidomain(forward_diamond(X, antidomain(antidomain(multiplication(Y, antidomain(backward_diamond(Y, X)))))))), multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X))))))
% 294.22/37.68 = { by lemma 53 R->L }
% 294.22/37.68 antidomain(domain_difference(antidomain(antidomain(antidomain(antidomain(forward_diamond(X, antidomain(antidomain(multiplication(Y, antidomain(backward_diamond(Y, X)))))))))), multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X))))))
% 294.22/37.68 = { by lemma 42 R->L }
% 294.22/37.68 antidomain(domain_difference(forward_diamond(antidomain(antidomain(forward_diamond(X, antidomain(antidomain(multiplication(Y, antidomain(backward_diamond(Y, X)))))))), one), multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X))))))
% 294.22/37.68 = { by lemma 56 }
% 294.22/37.68 antidomain(domain_difference(forward_diamond(antidomain(forward_box(X, antidomain(multiplication(Y, antidomain(backward_diamond(Y, X)))))), one), multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X))))))
% 294.22/37.68 = { by lemma 53 R->L }
% 294.22/37.68 antidomain(domain_difference(forward_diamond(antidomain(antidomain(antidomain(forward_box(X, antidomain(multiplication(Y, antidomain(backward_diamond(Y, X)))))))), one), multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X))))))
% 294.22/37.68 = { by lemma 42 }
% 294.22/37.68 antidomain(domain_difference(antidomain(antidomain(antidomain(antidomain(antidomain(forward_box(X, antidomain(multiplication(Y, antidomain(backward_diamond(Y, X)))))))))), multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X))))))
% 294.22/37.68 = { by lemma 53 }
% 294.22/37.68 antidomain(domain_difference(antidomain(antidomain(antidomain(forward_box(X, antidomain(multiplication(Y, antidomain(backward_diamond(Y, X)))))))), multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X))))))
% 294.22/37.68 = { by lemma 53 }
% 294.22/37.68 antidomain(domain_difference(antidomain(forward_box(X, antidomain(multiplication(Y, antidomain(backward_diamond(Y, X)))))), multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X))))))
% 294.22/37.68 = { by lemma 59 R->L }
% 294.22/37.68 antidomain(multiplication(antidomain(antidomain(antidomain(forward_box(X, antidomain(multiplication(Y, antidomain(backward_diamond(Y, X)))))))), antidomain(multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X)))))))
% 294.22/37.68 = { by lemma 53 }
% 294.22/37.68 antidomain(multiplication(antidomain(forward_box(X, antidomain(multiplication(Y, antidomain(backward_diamond(Y, X)))))), antidomain(multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X)))))))
% 294.22/37.68 = { by lemma 66 R->L }
% 294.22/37.68 antidomain(multiplication(antidomain(antidomain(multiplication(X, antidomain(antidomain(multiplication(Y, antidomain(backward_diamond(Y, X)))))))), antidomain(multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X)))))))
% 294.22/37.68 = { by axiom 24 (domain2) R->L }
% 294.22/37.68 antidomain(multiplication(antidomain(addition(antidomain(multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X))))), antidomain(multiplication(X, antidomain(antidomain(multiplication(Y, antidomain(backward_diamond(Y, X))))))))), antidomain(multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X)))))))
% 294.22/37.68 = { by lemma 66 }
% 294.22/37.68 antidomain(multiplication(antidomain(addition(antidomain(multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X))))), forward_box(X, antidomain(multiplication(Y, antidomain(backward_diamond(Y, X))))))), antidomain(multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X)))))))
% 294.22/37.68 = { by lemma 49 R->L }
% 294.22/37.68 antidomain(multiplication(antidomain(addition(antidomain(multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X))))), forward_box(X, antidomain(multiplication(Y, antidomain(backward_diamond(Y, X))))))), addition(antidomain(multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X))))), addition(antidomain(multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X))))), forward_box(X, antidomain(multiplication(Y, antidomain(backward_diamond(Y, X)))))))))
% 294.22/37.68 = { by lemma 44 }
% 294.22/37.68 antidomain(multiplication(antidomain(addition(antidomain(multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X))))), forward_box(X, antidomain(multiplication(Y, antidomain(backward_diamond(Y, X))))))), addition(antidomain(multiplication(X, multiplication(Y, antidomain(backward_diamond(Y, X))))), forward_box(X, antidomain(multiplication(Y, antidomain(backward_diamond(Y, X))))))))
% 294.22/37.68 = { by axiom 11 (domain1) }
% 294.22/37.68 antidomain(zero)
% 294.22/37.68 = { by lemma 31 }
% 294.22/37.68 one
% 294.22/37.68
% 294.22/37.68 Lemma 79: codomain(addition(domain(X), forward_box(Y, backward_diamond(Y, domain(X))))) = forward_box(Y, backward_diamond(Y, domain(X))).
% 294.22/37.68 Proof:
% 294.22/37.68 codomain(addition(domain(X), forward_box(Y, backward_diamond(Y, domain(X)))))
% 294.22/37.68 = { by lemma 43 R->L }
% 294.22/37.68 backward_diamond(addition(domain(X), forward_box(Y, backward_diamond(Y, domain(X)))), one)
% 294.22/37.69 = { by lemma 78 R->L }
% 294.22/37.69 backward_diamond(addition(domain(X), forward_box(Y, backward_diamond(Y, domain(X)))), forward_box(domain(X), forward_box(Y, backward_diamond(Y, domain(X)))))
% 294.22/37.69 = { by lemma 71 R->L }
% 294.22/37.69 backward_diamond(addition(domain(X), forward_box(Y, backward_diamond(Y, domain(X)))), antidomain(domain_difference(domain(X), forward_box(Y, backward_diamond(Y, domain(X))))))
% 294.22/37.69 = { by axiom 8 (multiplicative_right_identity) R->L }
% 294.22/37.69 backward_diamond(multiplication(addition(domain(X), forward_box(Y, backward_diamond(Y, domain(X)))), one), antidomain(domain_difference(domain(X), forward_box(Y, backward_diamond(Y, domain(X))))))
% 294.22/37.69 = { by lemma 63 R->L }
% 294.22/37.69 backward_diamond(multiplication(addition(domain(X), forward_box(Y, backward_diamond(Y, domain(X)))), addition(forward_box(Y, backward_diamond(Y, domain(X))), antidomain(forward_box(Y, backward_diamond(Y, domain(X)))))), antidomain(domain_difference(domain(X), forward_box(Y, backward_diamond(Y, domain(X))))))
% 294.22/37.69 = { by axiom 22 (right_distributivity) }
% 294.22/37.69 backward_diamond(addition(multiplication(addition(domain(X), forward_box(Y, backward_diamond(Y, domain(X)))), forward_box(Y, backward_diamond(Y, domain(X)))), multiplication(addition(domain(X), forward_box(Y, backward_diamond(Y, domain(X)))), antidomain(forward_box(Y, backward_diamond(Y, domain(X)))))), antidomain(domain_difference(domain(X), forward_box(Y, backward_diamond(Y, domain(X))))))
% 294.22/37.69 = { by lemma 50 R->L }
% 294.22/37.69 backward_diamond(addition(multiplication(addition(addition(domain(X), forward_box(Y, backward_diamond(Y, domain(X)))), antidomain(forward_box(Y, backward_diamond(Y, domain(X))))), forward_box(Y, backward_diamond(Y, domain(X)))), multiplication(addition(domain(X), forward_box(Y, backward_diamond(Y, domain(X)))), antidomain(forward_box(Y, backward_diamond(Y, domain(X)))))), antidomain(domain_difference(domain(X), forward_box(Y, backward_diamond(Y, domain(X))))))
% 294.22/37.69 = { by axiom 5 (additive_commutativity) R->L }
% 294.22/37.69 backward_diamond(addition(multiplication(addition(antidomain(forward_box(Y, backward_diamond(Y, domain(X)))), addition(domain(X), forward_box(Y, backward_diamond(Y, domain(X))))), forward_box(Y, backward_diamond(Y, domain(X)))), multiplication(addition(domain(X), forward_box(Y, backward_diamond(Y, domain(X)))), antidomain(forward_box(Y, backward_diamond(Y, domain(X)))))), antidomain(domain_difference(domain(X), forward_box(Y, backward_diamond(Y, domain(X))))))
% 294.22/37.69 = { by lemma 62 }
% 294.22/37.69 backward_diamond(addition(multiplication(addition(domain(X), addition(forward_box(Y, backward_diamond(Y, domain(X))), antidomain(forward_box(Y, backward_diamond(Y, domain(X)))))), forward_box(Y, backward_diamond(Y, domain(X)))), multiplication(addition(domain(X), forward_box(Y, backward_diamond(Y, domain(X)))), antidomain(forward_box(Y, backward_diamond(Y, domain(X)))))), antidomain(domain_difference(domain(X), forward_box(Y, backward_diamond(Y, domain(X))))))
% 294.22/37.69 = { by lemma 63 }
% 294.22/37.69 backward_diamond(addition(multiplication(addition(domain(X), one), forward_box(Y, backward_diamond(Y, domain(X)))), multiplication(addition(domain(X), forward_box(Y, backward_diamond(Y, domain(X)))), antidomain(forward_box(Y, backward_diamond(Y, domain(X)))))), antidomain(domain_difference(domain(X), forward_box(Y, backward_diamond(Y, domain(X))))))
% 294.22/37.69 = { by lemma 47 }
% 294.22/37.69 backward_diamond(addition(multiplication(one, forward_box(Y, backward_diamond(Y, domain(X)))), multiplication(addition(domain(X), forward_box(Y, backward_diamond(Y, domain(X)))), antidomain(forward_box(Y, backward_diamond(Y, domain(X)))))), antidomain(domain_difference(domain(X), forward_box(Y, backward_diamond(Y, domain(X))))))
% 294.22/37.69 = { by axiom 10 (multiplicative_left_identity) }
% 294.22/37.69 backward_diamond(addition(forward_box(Y, backward_diamond(Y, domain(X))), multiplication(addition(domain(X), forward_box(Y, backward_diamond(Y, domain(X)))), antidomain(forward_box(Y, backward_diamond(Y, domain(X)))))), antidomain(domain_difference(domain(X), forward_box(Y, backward_diamond(Y, domain(X))))))
% 294.22/37.69 = { by lemma 70 R->L }
% 294.22/37.69 backward_diamond(addition(forward_box(Y, backward_diamond(Y, domain(X))), multiplication(addition(domain(X), antidomain(antidomain(forward_box(Y, backward_diamond(Y, domain(X)))))), antidomain(forward_box(Y, backward_diamond(Y, domain(X)))))), antidomain(domain_difference(domain(X), forward_box(Y, backward_diamond(Y, domain(X))))))
% 294.22/37.69 = { by lemma 50 }
% 294.22/37.69 backward_diamond(addition(forward_box(Y, backward_diamond(Y, domain(X))), multiplication(domain(X), antidomain(forward_box(Y, backward_diamond(Y, domain(X)))))), antidomain(domain_difference(domain(X), forward_box(Y, backward_diamond(Y, domain(X))))))
% 294.22/37.69 = { by axiom 1 (domain4) }
% 294.22/37.69 backward_diamond(addition(forward_box(Y, backward_diamond(Y, domain(X))), multiplication(antidomain(antidomain(X)), antidomain(forward_box(Y, backward_diamond(Y, domain(X)))))), antidomain(domain_difference(domain(X), forward_box(Y, backward_diamond(Y, domain(X))))))
% 294.22/37.69 = { by lemma 59 }
% 294.22/37.69 backward_diamond(addition(forward_box(Y, backward_diamond(Y, domain(X))), domain_difference(X, forward_box(Y, backward_diamond(Y, domain(X))))), antidomain(domain_difference(domain(X), forward_box(Y, backward_diamond(Y, domain(X))))))
% 294.22/37.69 = { by lemma 60 R->L }
% 294.22/37.69 backward_diamond(addition(forward_box(Y, backward_diamond(Y, domain(X))), domain_difference(domain(X), forward_box(Y, backward_diamond(Y, domain(X))))), antidomain(domain_difference(domain(X), forward_box(Y, backward_diamond(Y, domain(X))))))
% 294.22/37.69 = { by axiom 5 (additive_commutativity) R->L }
% 294.22/37.69 backward_diamond(addition(domain_difference(domain(X), forward_box(Y, backward_diamond(Y, domain(X)))), forward_box(Y, backward_diamond(Y, domain(X)))), antidomain(domain_difference(domain(X), forward_box(Y, backward_diamond(Y, domain(X))))))
% 294.22/37.69 = { by axiom 16 (backward_diamond) }
% 294.22/37.69 codomain(multiplication(codomain(antidomain(domain_difference(domain(X), forward_box(Y, backward_diamond(Y, domain(X)))))), addition(domain_difference(domain(X), forward_box(Y, backward_diamond(Y, domain(X)))), forward_box(Y, backward_diamond(Y, domain(X))))))
% 294.22/37.69 = { by lemma 73 }
% 294.22/37.69 codomain(multiplication(codomain(antidomain(domain_difference(domain(X), forward_box(Y, backward_diamond(Y, domain(X)))))), forward_box(Y, backward_diamond(Y, domain(X)))))
% 294.22/37.69 = { by axiom 16 (backward_diamond) R->L }
% 294.22/37.69 backward_diamond(forward_box(Y, backward_diamond(Y, domain(X))), antidomain(domain_difference(domain(X), forward_box(Y, backward_diamond(Y, domain(X))))))
% 294.22/37.69 = { by lemma 71 }
% 294.22/37.69 backward_diamond(forward_box(Y, backward_diamond(Y, domain(X))), forward_box(domain(X), forward_box(Y, backward_diamond(Y, domain(X)))))
% 294.22/37.69 = { by lemma 78 }
% 294.22/37.69 backward_diamond(forward_box(Y, backward_diamond(Y, domain(X))), one)
% 294.22/37.69 = { by lemma 43 }
% 294.22/37.69 codomain(forward_box(Y, backward_diamond(Y, domain(X))))
% 294.22/37.69 = { by lemma 76 R->L }
% 294.22/37.69 codomain(coantidomain(antidomain(forward_box(Y, backward_diamond(Y, domain(X))))))
% 294.22/37.69 = { by lemma 40 }
% 294.22/37.69 coantidomain(antidomain(forward_box(Y, backward_diamond(Y, domain(X)))))
% 294.22/37.69 = { by lemma 76 }
% 294.22/37.69 forward_box(Y, backward_diamond(Y, domain(X)))
% 294.22/37.69
% 294.22/37.69 Goal 1 (goals): addition(domain(x0), forward_box(x1, backward_diamond(x1, domain(x0)))) = forward_box(x1, backward_diamond(x1, domain(x0))).
% 294.22/37.69 Proof:
% 294.22/37.69 addition(domain(x0), forward_box(x1, backward_diamond(x1, domain(x0))))
% 294.22/37.69 = { by axiom 4 (additive_idempotence) R->L }
% 294.22/37.69 addition(domain(x0), addition(forward_box(x1, backward_diamond(x1, domain(x0))), forward_box(x1, backward_diamond(x1, domain(x0)))))
% 294.22/37.69 = { by lemma 61 R->L }
% 294.22/37.69 addition(forward_box(x1, backward_diamond(x1, domain(x0))), addition(domain(x0), forward_box(x1, backward_diamond(x1, domain(x0)))))
% 294.22/37.69 = { by axiom 5 (additive_commutativity) }
% 294.22/37.69 addition(addition(domain(x0), forward_box(x1, backward_diamond(x1, domain(x0)))), forward_box(x1, backward_diamond(x1, domain(x0))))
% 294.22/37.69 = { by lemma 79 R->L }
% 294.22/37.69 addition(addition(domain(x0), forward_box(x1, backward_diamond(x1, domain(x0)))), codomain(addition(domain(x0), forward_box(x1, backward_diamond(x1, domain(x0))))))
% 294.22/37.69 = { by axiom 5 (additive_commutativity) R->L }
% 294.22/37.69 addition(codomain(addition(domain(x0), forward_box(x1, backward_diamond(x1, domain(x0))))), addition(domain(x0), forward_box(x1, backward_diamond(x1, domain(x0)))))
% 294.22/37.69 = { by lemma 39 R->L }
% 294.22/37.69 addition(codomain(addition(domain(x0), forward_box(x1, backward_diamond(x1, domain(x0))))), multiplication(addition(domain(x0), forward_box(x1, backward_diamond(x1, domain(x0)))), codomain(addition(domain(x0), forward_box(x1, backward_diamond(x1, domain(x0)))))))
% 294.22/37.69 = { by lemma 65 R->L }
% 294.22/37.69 multiplication(addition(addition(domain(x0), forward_box(x1, backward_diamond(x1, domain(x0)))), one), codomain(addition(domain(x0), forward_box(x1, backward_diamond(x1, domain(x0))))))
% 294.22/37.69 = { by axiom 5 (additive_commutativity) R->L }
% 294.22/37.69 multiplication(addition(one, addition(domain(x0), forward_box(x1, backward_diamond(x1, domain(x0))))), codomain(addition(domain(x0), forward_box(x1, backward_diamond(x1, domain(x0))))))
% 294.22/37.69 = { by lemma 62 }
% 294.22/37.69 multiplication(addition(domain(x0), addition(forward_box(x1, backward_diamond(x1, domain(x0))), one)), codomain(addition(domain(x0), forward_box(x1, backward_diamond(x1, domain(x0))))))
% 294.22/37.69 = { by lemma 64 }
% 294.22/37.69 multiplication(addition(domain(x0), one), codomain(addition(domain(x0), forward_box(x1, backward_diamond(x1, domain(x0))))))
% 294.22/37.69 = { by lemma 47 }
% 294.22/37.69 multiplication(one, codomain(addition(domain(x0), forward_box(x1, backward_diamond(x1, domain(x0))))))
% 294.22/37.69 = { by axiom 10 (multiplicative_left_identity) }
% 294.22/37.69 codomain(addition(domain(x0), forward_box(x1, backward_diamond(x1, domain(x0)))))
% 294.22/37.69 = { by lemma 79 }
% 294.22/37.69 forward_box(x1, backward_diamond(x1, domain(x0)))
% 294.22/37.69 % SZS output end Proof
% 294.22/37.69
% 294.22/37.69 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------