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;

theorem Th72:
  for X0, X1, X2 being non empty SubSpace of X st X1 is SubSpace
of X0 & X2 is SubSpace of X1 for g being Function of X0,Y holds (g|X1)|X2 = g|
  X2
proof
  let X0, X1, X2 be non empty SubSpace of X such that
A1: X1 is SubSpace of X0 and
A2: X2 is SubSpace of X1;
A3: X2 is SubSpace of X0 by A1,A2,TSEP_1:7;
  let g be Function of X0,Y;
  set h = g|X1;
A4: h = g|(the carrier of X1) & the carrier of X2 c= the carrier of X1 by A1,A2
,Def5,TSEP_1:4;
  thus h|X2 = h|(the carrier of X2) by A2,Def5
    .= g|(the carrier of X2) by A4,FUNCT_1:51
    .= g|X2 by A3,Def5;
end;
