
theorem Th1:
  for x,y,z,a,b,c being set st a <> b & b <> c & c <> a ex f being
  Function st f.a = x & f.b = y & f.c = z
proof
  let x,y,z,a,b,c be set such that
A1: a <> b and
A2: b <> c and
A3: c <> a;
  set fb = b.-->y;
A5: a nin dom fb by A1,TARSKI:def 1;
A6: b in dom fb by TARSKI:def 1;
  set fc = c.-->z;
  set fa = a.-->x;
  take f = fa+*fb+*fc;
  a nin dom fc by A3,TARSKI:def 1;
  hence f.a = (fa+*fb).a by FUNCT_4:11
    .= fa.a by A5,FUNCT_4:11
    .= x by FUNCOP_1:72;
  b nin dom fc by A2,TARSKI:def 1;
  hence f.b = (fa+*fb).b by FUNCT_4:11
    .= fb.b by A6,FUNCT_4:13
    .= y by FUNCOP_1:72;
  c in dom fc by TARSKI:def 1;
  hence f.c = fc.c by FUNCT_4:13
    .= z by FUNCOP_1:72;
end;
