theorem     :: REALSET1:4
  A c= X implies [|A,A|] c= [|X,X|]
proof
  assume
A1: A c= X;
  let i;
  assume
A2: i in I;
  then A.i c= X.i by A1;
  then [:A.i,A.i:] c= [:X.i,X.i:] by ZFMISC_1:96;
  then [|A,A|].i c= [:X.i,X.i:] by A2,PBOOLE:def 16;
  hence thesis by A2,PBOOLE:def 16;
end;
