theorem Th18:
  X |-| Y iff X |- Y & Y |- X
proof
  thus X |-| Y implies X |- Y & Y |- X by Th1;
  assume that
A1: X |- Y and
A2: Y |- X;
    let p;
A3: now
      assume Y |- p;
      then Y |- {p} by Th10;
      then X |- {p} by A1,Th9;
      hence X |- p by Th10;
    end;
    now
      assume X |- p;
      then X |- {p} by Th10;
      then Y |- {p} by A2,Th9;
      hence Y |- p by Th10;
    end;
    hence X |- p iff Y |- p by A3;
  end;
