let ti be Function of X, INT ; :: thesis: ( ti = t1 - t2 iff for s being Element of X holds ti . s = (t1 . s) - (t2 . s) )
A1: ( dom (t1 - t2) = (dom t1) /\ (dom t2) & dom t1 = X & dom t2 = X & dom ti = X ) by FUNCT_2:def 1, VALUED_1:12;
thus ( ti = t1 - t2 implies for s being Element of X holds ti . s = (t1 . s) - (t2 . s) ) by VALUED_1:15; :: thesis: ( ( for s being Element of X holds ti . s = (t1 . s) - (t2 . s) ) implies ti = t1 - t2 )
assume A2: for s being Element of X holds ti . s = (t1 . s) - (t2 . s) ; :: thesis: ti = t1 - t2

now

let s be set ; :: thesis: ( s in X implies ti . s = (t1 - t2) . s )
assume A3: s in X ; :: thesis: ti . s = (t1 - t2) . s
hence ti . s = (t1 . s) - (t2 . s) by A2
.= (t1 - t2) . s by A3, VALUED_1:15 ;
:: thesis: verum

end;

hence ti = t1 - t2 by A1, FUNCT_1:9; :: thesis: verum