reserve a,b,c,d for Real;

theorem
  for X, Y being non empty SubSpace of R^1, f being continuous Function of X,Y
  holds
  (ex a,b being Real st a <= b & [.a,b.] c= the carrier of X &
     f.:[.a,b.] c= [.a,b.]) implies
  ex x being Point of X st f.x = x
proof
  let X, Y be non empty SubSpace of R^1, f be continuous Function of X,Y;
  given a,b being Real such that
A1: a <= b and
A2: [.a,b.] c= the carrier of X and
A3: f.:[.a,b.] c= [.a,b.];
  set g = (Y incl R^1) * f;
  the carrier of Y c= the carrier of R^1 by BORSUK_1:1;
  then reconsider B = f.:[.a,b.] as Subset of R^1 by XBOOLE_1:1;
  g.:[.a,b.] = (Y incl R^1).:(f.:[.a,b.]) by RELAT_1:126;
  then g.:[.a,b.] = ((id R^1)|Y).:B by TMAP_1:def 6;
  then g.:[.a,b.] = (id R^1).:B by TMAP_1:55;
  then
A4: g.:[.a,b.] c= [.a,b.] by A3,FUNCT_1:92;
A5: (Y incl R^1) is continuous Function of Y,R^1 & R^1 is SubSpace of R^1 by
TMAP_1:87,TSEP_1:2;
  the carrier of X c= the carrier of R^1 by BORSUK_1:1;
  then
A6: [.a,b.] c= the carrier of R^1 by A2;
  now
    consider x being Point of X such that
A7: g.x = x by A1,A2,A5,A6,A4,Th23;
    the carrier of Y c= the carrier of R^1 by BORSUK_1:1;
    then reconsider y = f.x as Point of R^1;
    take x;
    thus f.x = (Y incl R^1).y by TMAP_1:84
      .= x by A7,FUNCT_2:15;
  end;
  hence thesis;
end;
