reserve a, r, s for Real;

theorem Th20:
  cLoop(r) = CircleMap * ExtendInt(r)
proof
  for x being Point of I[01] holds (cLoop(r)).x = (CircleMap * ExtendInt(r
  )). x
  proof
    let x be Point of I[01];
A1: (ExtendInt(r)).x = r*x by Def1;
    thus (cLoop(r)).x = |[ cos(2*PI*r*x), sin(2*PI*r*x) ]| by Def4
      .= |[ cos(2*PI*(ExtendInt(r)).x), sin(2*PI*(ExtendInt(r)).x) ]| by A1
      .= CircleMap.((ExtendInt(r)).x) by TOPREALB:def 11
      .= (CircleMap * ExtendInt(r)).x by FUNCT_2:15;
  end;
  hence thesis;
end;
