
theorem Th22:
  for X, Y being TopSpace, Z being Subset of [:Y, X:], V being
Subset of X, W being Subset of Y st Z = [:W, V:] holds the TopStruct of [:Y | W
  , X | V:] = the TopStruct of [:Y, X:] | Z
proof
  let X, Y be TopSpace, Z be Subset of [:Y, X:], V be Subset of X, W be Subset
  of Y;
  reconsider A = [:Y | W, X | V:] as SubSpace of [:Y, X:] by Th21;
  assume
A1: Z = [:W, V:];
  the carrier of A = [:the carrier of (Y|W), the carrier of (X|V):] by
BORSUK_1:def 2
    .= [:the carrier of (Y|W), V:] by PRE_TOPC:8
    .= Z by A1,PRE_TOPC:8
    .= the carrier of ([:Y, X:]|Z) by PRE_TOPC:8;
  then A is SubSpace of [:Y, X:] | Z & [:Y, X:] | Z is SubSpace of A by
TOPMETR:3;
  hence thesis by TSEP_1:6;
end;
