theorem Th20:
  X |-| Y iff Cn(X) = Cn(Y)
proof
A1: now
    assume
A2: X |-| Y;
    then Y |- X by Th18;
    then X c= Cn(Y) by Th7;
    then
A3: Cn(X) c= Cn(Y) by CQC_THE1:15,16;
    X |- Y by A2,Th18;
    then Y c= Cn(X) by Th7;
    then Cn(Y) c= Cn(X) by CQC_THE1:15,16;
    hence Cn(X) = Cn(Y) by A3,XBOOLE_0:def 10;
  end;
  now
    assume
A4: Cn(X) = Cn(Y);
    X c= Cn(X) by CQC_THE1:17;
    then
A5: Y |- X by A4,Th7;
    Y c= Cn(Y) by CQC_THE1:17;
    then X |- Y by A4,Th7;
    hence X |-| Y by A5,Th18;
  end;
  hence thesis by A1;
end;
