%   ORIGINAL: 'h4/thm/quotient_sum/SUM_MAP_PRS_'
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Goal: !R1 abs1 rep1. 'h4/const/quotient/QUOTIENT' R1 abs1 rep1 ==> (!R2 abs2 rep2. 'h4/const/quotient/QUOTIENT' R2 abs2 rep2 ==> (!R3 abs3 rep3. 'h4/const/quotient/QUOTIENT' R3 abs3 rep3 ==> (!R4 abs4 rep4. 'h4/const/quotient/QUOTIENT' R4 abs4 rep4 ==> (!f g. 'h4/const/sum/++' f g = 'h4/const/quotient/-->' ('h4/const/sum/++' rep1 rep3) ('h4/const/sum/++' abs2 abs4) ('h4/const/sum/++' ('h4/const/quotient/-->' abs1 rep2 f) ('h4/const/quotient/-->' abs3 rep4 g))))))
%   PROCESSED
% Assm ['HL_TRUTH']: T
% Assm ['HL_FALSITY']: ~F
% Assm ['HL_BOOL_CASES']: !t. (t <=> T) \/ (t <=> F)
% Assm ['HL_EXT']: !f g. (!x. happ f x = happ g x) ==> f = g
% Goal: !R1 abs1 rep1. 'h4/const/quotient/QUOTIENT' R1 abs1 rep1 ==> (!R2 abs2 rep2. 'h4/const/quotient/QUOTIENT' R2 abs2 rep2 ==> (!R3 abs3 rep3. 'h4/const/quotient/QUOTIENT' R3 abs3 rep3 ==> (!R4 abs4 rep4. 'h4/const/quotient/QUOTIENT' R4 abs4 rep4 ==> (!f g. 'h4/const/sum/++' f g = 'h4/const/quotient/-->' ('h4/const/sum/++' rep1 rep3) ('h4/const/sum/++' abs2 abs4) ('h4/const/sum/++' ('h4/const/quotient/-->' abs1 rep2 f) ('h4/const/quotient/-->' abs3 rep4 g))))))
fof('HL_TRUTH', axiom, p(s(bool,'T'))).
fof('HL_FALSITY', axiom, ~ (p(s(bool,'F')))).
fof('HL_BOOL_CASES', axiom, ![T]: (s(bool,T) = s(bool,'T') | s(bool,T) = s(bool,'F'))).
fof('HL_EXT', axiom, ![V_3f14569,V_3f14565]: ![F0, G0]: (![X]: s(V_3f14565,happ(s(fun(V_3f14569,V_3f14565),F0),s(V_3f14569,X))) = s(V_3f14565,happ(s(fun(V_3f14569,V_3f14565),G0),s(V_3f14569,X))) => s(fun(V_3f14569,V_3f14565),F0) = s(fun(V_3f14569,V_3f14565),G0))).
fof('h4/thm/quotient_sum/SUM_MAP_PRS_', conjecture, ![A,B,E,F,C,D,G,H]: ![R1, Abs1, Rep1]: (p(s(bool,'h4/const/quotient/QUOTIENT'(s(fun(A,fun(A,bool)),R1),s(fun(A,E),Abs1),s(fun(E,A),Rep1)))) => ![R2, Abs2, Rep2]: (p(s(bool,'h4/const/quotient/QUOTIENT'(s(fun(B,fun(B,bool)),R2),s(fun(B,F),Abs2),s(fun(F,B),Rep2)))) => ![R3, Abs3, Rep3]: (p(s(bool,'h4/const/quotient/QUOTIENT'(s(fun(C,fun(C,bool)),R3),s(fun(C,G),Abs3),s(fun(G,C),Rep3)))) => ![R4, Abs4, Rep4]: (p(s(bool,'h4/const/quotient/QUOTIENT'(s(fun(D,fun(D,bool)),R4),s(fun(D,H),Abs4),s(fun(H,D),Rep4)))) => ![F0, G0]: s(fun('h4/type/sum/sum'(E,G),'h4/type/sum/sum'(F,H)),'h4/const/sum/++'(s(fun(E,F),F0),s(fun(G,H),G0))) = s(fun('h4/type/sum/sum'(E,G),'h4/type/sum/sum'(F,H)),'h4/const/quotient/-->'(s(fun('h4/type/sum/sum'(E,G),'h4/type/sum/sum'(A,C)),'h4/const/sum/++'(s(fun(E,A),Rep1),s(fun(G,C),Rep3))),s(fun('h4/type/sum/sum'(B,D),'h4/type/sum/sum'(F,H)),'h4/const/sum/++'(s(fun(B,F),Abs2),s(fun(D,H),Abs4))),s(fun('h4/type/sum/sum'(A,C),'h4/type/sum/sum'(B,D)),'h4/const/sum/++'(s(fun(A,B),'h4/const/quotient/-->'(s(fun(A,E),Abs1),s(fun(F,B),Rep2),s(fun(E,F),F0))),s(fun(C,D),'h4/const/quotient/-->'(s(fun(C,G),Abs3),s(fun(H,D),Rep4),s(fun(G,H),G0)))))))))))).
