reserve i for object, I for set,
  f for Function,
  x, x1, x2, y, A, B, X, Y, Z for ManySortedSet of I;

theorem     :: PARTFUN1:1
  x c= [|A,B|] & y c= [|X,Y|] implies x (\/) y c= [|A (\/) X,B (\/) Y|]
proof
  assume that
A1: x c= [|A,B|] and
A2: y c= [|X,Y|];
  let i;
  assume
A3: i in I;
  then
A4: x.i c= [|A,B|].i by A1;
A5: y.i c= [|X,Y|].i by A2,A3;
A6: x.i c= [:A.i,B.i:] by A3,A4,PBOOLE:def 16;
  y.i c= [:X.i,Y.i:] by A3,A5,PBOOLE:def 16;
  then x.i \/ y.i c= [:A.i \/ X.i,B.i \/ Y.i:] by A6,ZFMISC_1:119;
  then (x (\/) y).i c= [:A.i \/ X.i,B.i \/ Y.i:] by A3,PBOOLE:def 4;
  then (x (\/) y).i c= [:(A (\/) X).i,B.i \/ Y.i:] by A3,PBOOLE:def 4;
  then (x (\/) y).i c= [:(A (\/) X).i,(B (\/) Y).i:] by A3,PBOOLE:def 4;
  hence thesis by A3,PBOOLE:def 16;
end;
