reserve p,q,x,x1,x2,y,y1,y2,z,z1,z2 for set;
reserve A,B,V,X,X1,X2,Y,Y1,Y2,Z for set;
reserve C,C1,C2,D,D1,D2 for non empty set;

theorem Th6:
  for f,g being Function st dom f = [:X,Y:] & dom g = [:X,Y:] &
  for x,y being object st x in X & y in Y holds f.(x,y) = g.(x,y)
    holds f = g
proof
  let f,g be Function such that
A1: dom f = [:X,Y:] & dom g = [:X,Y:] and
A2: for x,y being object st x in X & y in Y holds f.(x,y) = g.(x,y);
  for p being object holds p in [:X,Y:] implies f.p = g.p
  proof let p be object;
    assume p in [:X,Y:];
    then consider x,y being object such that
A3: x in X & y in Y and
A4: p = [x,y] by ZFMISC_1:def 2;
    f.(x,y) = g.(x,y) by A2,A3;
    hence thesis by A4;
  end;
  hence thesis by A1;
end;
