let f1, f2 be Function of (TOP-REAL 2),R^1; :: thesis: ( ( for p being Point of (TOP-REAL 2) holds f1 . p = tricord3 (p1,p2,p3,p) ) & ( for p being Point of (TOP-REAL 2) holds f2 . p = tricord3 (p1,p2,p3,p) ) implies f1 = f2 )
assume that
A4: for p being Point of (TOP-REAL 2) holds f1 . p = tricord3 (p1,p2,p3,p) and
A5: for p being Point of (TOP-REAL 2) holds f2 . p = tricord3 (p1,p2,p3,p) ; :: thesis: f1 = f2
A6: for x being object st x in dom f1 holds
f1 . x = f2 . x
proof
let x be object ; :: thesis: ( x in dom f1 implies f1 . x = f2 . x )
assume x in dom f1 ; :: thesis: f1 . x = f2 . x
then reconsider p0 = x as Point of (TOP-REAL 2) by FUNCT_2:def 1;
f1 . p0 = tricord3 (p1,p2,p3,p0) by A4;
hence f1 . x = f2 . x by A5; :: thesis: verum
end;
dom f1 = the carrier of (TOP-REAL 2) by FUNCT_2:def 1;
then dom f1 = dom f2 by FUNCT_2:def 1;
hence f1 = f2 by A6, FUNCT_1:2; :: thesis: verum