
theorem Th7:
  for X,Y,X1,Y1 being TopSpace st the TopStruct of X = the
TopStruct of X1 & the TopStruct of Y = the TopStruct of Y1 holds [:X,Y:] = [:X1
  ,Y1:]
proof
  let X,Y,X1,Y1 be TopSpace;
  assume
A1: the TopStruct of X = the TopStruct of X1 & the TopStruct of Y = the
  TopStruct of Y1;
  set U2 = {union A where A is Subset-Family of [:X1,Y1:]: A c= { [:X2,Y2:]
where X2 is Subset of X1, Y2 is Subset of Y1 : X2 in the topology of X1 & Y2 in
  the topology of Y1}};
A2: the carrier of [:X,Y:] = [:the carrier of X,the carrier of Y:] by
BORSUK_1:def 2
    .= the carrier of [:X1,Y1:] by A1,BORSUK_1:def 2;
  then the topology of [:X,Y:] = U2 by A1,BORSUK_1:def 2
    .= the topology of [:X1,Y1:] by BORSUK_1:def 2;
  hence thesis by A2;
end;
