theorem Th20: Q in compn P implies Q is complete
  proof
    assume
A1: Q in compn P;
    then consider Q1 be PNPair such that
    Q1 = Q and
A2: Q1 in comp untn P;
    consider x such that
    Q1 in x and
A3: x in {comp R where R is PNPair : R in untn P} by A2,TARSKI:def 4;
    ex R be PNPair st x = comp R & R in untn P by A3;
    hence Q is complete by Th19,A1,Th13;
  end;
