reserve Y for TopStruct;
reserve X for non empty TopSpace;
reserve X for almost_discrete non empty TopSpace;
reserve X,Y for non empty TopSpace;
reserve X for discrete non empty TopSpace,
  X0 for non empty SubSpace of X;
reserve X for almost_discrete non empty TopSpace,
  X0 for maximal_discrete non empty SubSpace of X;
reserve X0 for discrete non empty SubSpace of X;

theorem Th72:
  ex r being continuous Function of X,X0 st r is being_a_retraction
proof
  consider Z0 being strict non empty SubSpace of X such that
A1: X0 is SubSpace of Z0 and
A2: Z0 is maximal_discrete by Th61;
  reconsider Z0 as maximal_discrete strict non empty SubSpace of X by A2;
  reconsider Z1 = Z0 as non empty TopSpace;
  reconsider Z1 as discrete non empty TopSpace;
  reconsider X1 = X0 as non empty SubSpace of Z1 by A1;
  consider g being continuous Function of Z1,X1 such that
A3: g is being_a_retraction by Th66;
  reconsider g as continuous Function of Z0,X0;
  consider h being continuous Function of X,Z0 such that
A4: h is being_a_retraction by Th68;
  reconsider r = g * h as continuous Function of X,X0;
  take r;
  for x being Point of X st x in the carrier of X0 holds r.x = x
  proof
    let x be Point of X;
    assume
A5: x in the carrier of X0;
    the carrier of X1 c= the carrier of Z1 by BORSUK_1:1;
    then reconsider y = x as Point of Z1 by A5;
    g.y = y by A3,A5,BORSUK_1:def 16;
    then g.(h.x) = x by A4,BORSUK_1:def 16;
    hence thesis by FUNCT_2:15;
  end;
  hence thesis by BORSUK_1:def 16;
end;
