let ti be INT-Expression of A,f; ( ti = t1 div t2 iff for s being Element of Funcs (X,INT) holds ti . s = (t1 . s) div (t2 . s) )
A1:
dom t1 = Funcs (X,INT)
by FUNCT_2:def 1;
A2:
dom t2 = Funcs (X,INT)
by FUNCT_2:def 1;
A3:
dom (t1 div t2) = (dom t1) /\ (dom t2)
by Def3;
hence
( ti = t1 div t2 implies for s being Element of Funcs (X,INT) holds ti . s = (t1 . s) div (t2 . s) )
by A1, A2, Def3; ( ( for s being Element of Funcs (X,INT) holds ti . s = (t1 . s) div (t2 . s) ) implies ti = t1 div t2 )
A4:
dom ti = Funcs (X,INT)
by FUNCT_2:def 1;
assume
for s being Element of Funcs (X,INT) holds ti . s = (t1 . s) div (t2 . s)
; ti = t1 div t2
then
for s being object st s in Funcs (X,INT) holds
ti . s = (t1 . s) div (t2 . s)
;
hence
ti = t1 div t2
by A3, A1, A2, A4, Def3; verum