reserve A, B for non empty set,
  A1, A2, A3 for non empty Subset of A;
reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X0, X1, X2, Y1, Y2 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace;
reserve f for Function of X,Y;
reserve X,Y,Z for non empty TopSpace;
reserve f for Function of X,Y,
  g for Function of Y,Z;
reserve X, Y for non empty TopSpace,
  X0 for non empty SubSpace of X;
reserve f for Function of X,Y;
reserve f for Function of X,Y,
  X0 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace,
  X0, X1 for non empty SubSpace of X;
reserve f for Function of X,Y,
  g for Function of X0,Y;
reserve X0, X1, X2 for non empty SubSpace of X;
reserve f for Function of X,Y,
  g for Function of X0,Y;
reserve X for non empty TopSpace,
  H, G for Subset of X;
reserve A for Subset of X;
reserve X0 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace;

theorem
  for X1, X2 being non empty SubSpace of X, f1 being Function of X1,Y,
  f2 being Function of X2,Y st f1|(X1 meet X2) = f2|(X1 meet X2) holds (X1 is
SubSpace of X2 iff f1 union f2 = f2) & (X2 is SubSpace of X1 iff f1 union f2 =
  f1)
proof
  let X1, X2 be non empty SubSpace of X, f1 be Function of X1,Y, f2 be
  Function of X2,Y;
  reconsider Y1 = X1, Y2 = X2, Y3 = X1 union X2 as SubSpace of X1 union X2 by
TSEP_1:2,22;
  assume
A1: f1|(X1 meet X2) = f2|(X1 meet X2);
A2: now
    assume X1 is SubSpace of X2;
    then
A3: the TopStruct of X2 = X1 union X2 by TSEP_1:23;
    (f1 union f2)|X2 = f2 by A1,Def12;
    then (f1 union f2)|the carrier of Y2 = f2 by Def5;
    then (f1 union f2)|the carrier of Y3 = f2 by A3;
    then (f1 union f2)|(X1 union X2) = f2 by Def5;
    hence f1 union f2 = f2 by Th67;
  end;
A4: now
    assume X2 is SubSpace of X1;
    then
A5: the TopStruct of X1 = X1 union X2 by TSEP_1:23;
    (f1 union f2)|X1 = f1 by A1,Def12;
    then (f1 union f2)|the carrier of Y1 = f1 by Def5;
    then (f1 union f2)|the carrier of Y3 = f1 by A5;
    then (f1 union f2)|(X1 union X2) = f1 by Def5;
    hence f1 union f2 = f1 by Th67;
  end;
  now
A6: dom (f1 union f2) = the carrier of X1 union X2 & dom f2 = the carrier
    of X2 by FUNCT_2:def 1;
    assume f1 union f2 = f2;
    then X1 union X2 = the TopStruct of X2 by A6,TSEP_1:5;
    hence X1 is SubSpace of X2 by TSEP_1:23;
  end;
  hence X1 is SubSpace of X2 iff f1 union f2 = f2 by A2;
  now
A7: dom (f1 union f2) = the carrier of X1 union X2 & dom f1 = the carrier
    of X1 by FUNCT_2:def 1;
    assume f1 union f2 = f1;
    then X1 union X2 = the TopStruct of X1 by A7,TSEP_1:5;
    hence X2 is SubSpace of X1 by TSEP_1:23;
  end;
  hence X2 is SubSpace of X1 iff f1 union f2 = f1 by A4;
end;
