let ti be INT-Expression of A,f; :: thesis: ( ti = eq t1,t2 iff for s being Element of Funcs X,INT holds ti . s = IFEQ (t1 . s),(t2 . s),1,0 )
A1: dom t1 = Funcs X,INT by FUNCT_2:def 1;
A2: dom t2 = Funcs X,INT by FUNCT_2:def 1;
A3: dom (eq t1,t2) = (dom t1) /\ (dom t2) by Def7;
hence ( ti = eq t1,t2 implies for s being Element of Funcs X,INT holds ti . s = IFEQ (t1 . s),(t2 . s),1,0 ) by A1, A2, Def7; :: thesis: ( ( for s being Element of Funcs X,INT holds ti . s = IFEQ (t1 . s),(t2 . s),1,0 ) implies ti = eq t1,t2 )
A4: dom ti = Funcs X,INT by FUNCT_2:def 1;
assume for s being Element of Funcs X,INT holds ti . s = IFEQ (t1 . s),(t2 . s),1,0 ; :: thesis: ti = eq t1,t2
then for s being set st s in Funcs X,INT holds
ti . s = IFEQ (t1 . s),(t2 . s),1,0 ;
hence ti = eq t1,t2 by A3, A1, A2, A4, Def7; :: thesis: verum