theorem Th21: A 'U' B in rng P`2 & Q in compn P implies untn(A,B) in rng Q`2
  proof
    assume that
A1: A 'U' B in rng P`2 and
A2: Q in compn P;
    consider Q1 be Element of pairs such that
    Q = Q1 and
A3: Q1 in comp untn P by A2;
    consider x such that
    Q1 in x and
A4: x in {(comp R) where R is Element of pairs : R in untn P}
    by TARSKI:def 4,A3;
    consider R be PNPair such that
    x = comp R and
A5: R in untn P by A4;
    ex R1 be PNPair st R1 = R & rng R1`1 = untn rng P`1
    & rng R1`2 = untn rng P`2 by A5;
    then A6: untn(A,B) in rng (R`2) by A1;
    Q is_completion_of R by A5,A2,Th19;
    then rng (R`2) c= rng (Q`2);
    hence untn(A,B) in rng Q`2 by A6;
  end;
