theorem ThC1:
  for s1,s2 being SortSymbol of S
  for x1 being Element of X.s1
  for x2 being Element of X.s2
  holds s1 <> s2 or x1 <> x2 iff x1-term is x2-omitting
  proof
    let s1,s2 be SortSymbol of S;
    let x1 be Element of X.s1;
    let x2 be Element of X.s2;
    hereby
      assume s1 <> s2 or x1 <> x2;
      then [x1,s1] <> [x2,s2] by XTUPLE_0:1;
      then [x1,s1] nin {[x2,s2]} by TARSKI:def 1;
      hence x1-term is x2-omitting by FUNCOP_1:16;
    end;
    assume Coim(x1-term,[x2,s2]) = {};
    then [x1,s1] nin {[x2,s2]} by FUNCOP_1:14;
    hence s1 <> s2 or x1 <> x2 by TARSKI:def 1;
  end;
