reserve a,b,p,x,x9,x1,x19,x2,y,y9,y1,y19,y2,z,z9,z1,z2 for object,
   X,X9,Y,Y9,Z,Z9 for set;
reserve A,D,D9 for non empty set;
reserve f,g,h for Function;

theorem Th63:
 for x1,x2,y1,y2 being object holds
  (x1 <> x2 implies ((x1,x2) --> (y1,y2)).x1 = y1) &
  ((x1,x2) --> (y1,y2)).x2 = y2
proof let x1,x2,y1,y2 be object;
  set f = {x1} --> y1, g = {x2} -->y2, h = (x1,x2) --> (y1,y2);
A1: x2 in {x2} by TARSKI:def 1;
A2: x1 in {x1} by TARSKI:def 1;
  hereby
    assume x1 <> x2;
    then not x1 in dom g by TARSKI:def 1;
    then h.x1 = f.x1 by Th11;
    hence h.x1 = y1 by A2,FUNCOP_1:7;
  end;
  {x2} = dom g;
  then h.x2 = g.x2 by A1,Th13;
  hence thesis by A1,FUNCOP_1:7;
end;
