 reserve x,y for Element of [.0,1.];

theorem
  #R 1 = AffineMap (1,0) | right_open_halfline 0
  proof
    set f = #R 1;
    set g = AffineMap (1,0) | right_open_halfline 0;
A1: dom f = right_open_halfline 0 by TAYLOR_1:def 4;
    dom AffineMap (1,0) = REAL by FUNCT_2:def 1; then
BA: dom f = dom g by A1,RELAT_1:62;
    reconsider p = 1 as Real;
    for x being object st x in dom f holds f.x = g.x
    proof
      let x be object;
      assume x in dom f; then
      reconsider xx = x as Element of right_open_halfline 0 by TAYLOR_1:def 4;
A2:   xx > 0 by XXREAL_1:4;
      f.x = xx #R p by TAYLOR_1:def 4 .= 1 * xx + 0 by PREPOWER:72,A2
         .= (AffineMap (1,0)).xx by FCONT_1:def 4
         .= g.x by FUNCT_1:49;
      hence thesis;
    end;
    hence thesis by BA,FUNCT_1:2;
  end;
