reserve I for set,
  x,x1,x2,y,z for set,
  A for non empty set;
reserve C,D for Category;
reserve a,b,c,d for Object of C;
reserve f,g,h,i,j,k,p1,p2,q1,q2,i1,i2,j1,j2 for Morphism of C;

theorem Th3:
  x1 <> x2 implies for y1,y2 being Element of A holds ((x1,x2) -->
  (y1,y2))/.x1 = y1 & ((x1,x2) --> (y1,y2))/.x2 = y2
proof
  assume
A1: x1 <> x2;
  let y1,y2 be Element of A;
  set h = (x1,x2) --> (y1,y2);
A2: h.x2 = y2 & x1 in {x1,x2} by FUNCT_4:63,TARSKI:def 2;
A3: x2 in {x1,x2} by TARSKI:def 2;
  h.x1 = y1 by A1,FUNCT_4:63;
  hence thesis by A2,A3,FUNCT_2:def 13;
end;
