reserve X for non empty 1-sorted;
reserve A, A1, A2, B1, B2 for Subset of X;
reserve X for non empty TopSpace;
reserve A, A1, A2, B1, B2 for Subset of X;
reserve X for non empty 1-sorted;
reserve A, A1, A2, B1, B2 for Subset of X;
reserve X for non empty TopSpace,
  A1, A2 for Subset of X;
reserve X0 for non empty SubSpace of X,
  B1, B2 for Subset of X0;
reserve X0, X1, X2, Y1, Y2 for non empty SubSpace of X;
reserve X for non empty TopSpace;
reserve X0 for non empty SubSpace of X;

theorem Th44:
  for X1, X2 being SubSpace of X, Y1, Y2 being SubSpace of X0 st
  the carrier of X1 = the carrier of Y1 & the carrier of X2 = the carrier of Y2
  holds X1,X2 are_separated iff Y1,Y2 are_separated
proof
  let X1, X2 be SubSpace of X, X01, X02 be SubSpace of X0;
  reconsider A1 = the carrier of X1, A2 = the carrier of X2 as Subset of X by
TSEP_1:1;
  reconsider B1 = the carrier of X01, B2 = the carrier of X02 as Subset of X0
  by TSEP_1:1;
  assume
A1: the carrier of X1 = the carrier of X01 & the carrier of X2 = the
  carrier of X02;
  thus X1,X2 are_separated implies X01,X02 are_separated
  proof
    assume X1,X2 are_separated;
    then A1,A2 are_separated;
    then for B1,B2 be Subset of X0 holds B1 = the carrier of X01 & B2 = the
    carrier of X02 implies B1,B2 are_separated by A1,Th25;
    hence thesis;
  end;
  thus X01,X02 are_separated implies X1,X2 are_separated
  proof
    assume X01,X02 are_separated;
    then B1,B2 are_separated;
    then for A1,A2 be Subset of X holds A1 = the carrier of X1 & A2 = the
    carrier of X2 implies A1,A2 are_separated by A1,Th25;
    hence thesis;
  end;
end;
