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

theorem     :: BORSUK_1:7
  [|x,X|] c= [|y,Y|] & [|x,X|] is non-empty implies x c= y & X c= Y
proof
  assume that
A1: [|x,X|] c= [|y,Y|] and
A2: [|x,X|] is non-empty;
  thus x c= y
  proof
    let i;
    assume
A3: i in I;
    then [|x,X|].i c= [|y,Y|].i by A1;
    then [:x.i,X.i:] c= [|y,Y|].i by A3,PBOOLE:def 16;
    then
A4: [:x.i,X.i:] c= [:y.i,Y.i:] by A3,PBOOLE:def 16;
    [|x,X|].i is non empty by A2,A3;
    then [:x.i,X.i:] is non empty by A3,PBOOLE:def 16;
    hence thesis by A4,ZFMISC_1:114;
  end;
  let i;
  assume
A5: i in I;
  then [|x,X|].i c= [|y,Y|].i by A1;
  then [:x.i,X.i:] c= [|y,Y|].i by A5,PBOOLE:def 16;
  then
A6: [:x.i,X.i:] c= [:y.i,Y.i:] by A5,PBOOLE:def 16;
  [|x,X|].i is non empty by A2,A5;
  then [:x.i,X.i:] is non empty by A5,PBOOLE:def 16;
  hence thesis by A6,ZFMISC_1:114;
end;
