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

theorem
  for u being Real st u in ].0,1.] holds
    ( #R x + AffineMap (-x, x-1)).u = u to_power x - 1 + x - x * u
  proof
      set f1 = #R x;
      set f2 = AffineMap (-x, x-1);
      reconsider Y = ].0,1.[ as open Subset of REAL;
      set f = f1 + f2;
      set A = right_open_halfline 0;
BX:   dom f = dom f1 /\ dom f2 by VALUED_1:def 1
           .= right_open_halfline 0 /\ dom f2 by TAYLOR_1:def 4
           .= right_open_halfline 0 /\ REAL by FUNCT_2:def 1
           .= right_open_halfline 0 by XBOOLE_1:28;
        let u be Real;
        assume
S1:     u in ].0,1.]; then
ZE:     u > 0 by XXREAL_1:2;
        1 < +infty by XXREAL_0:9,XREAL_0:def 1; then
h0:     ].0,1.] c= ].0,+infty.[ by XXREAL_1:49; then
        f.u = f1.u + f2.u by S1,BX,VALUED_1:def 1
           .= (u #R x) + f2.u by TAYLOR_1:def 4,h0,S1
           .= (u to_power x) + f2.u by ZE,POWER:def 2
           .= (u to_power x) + ((-x)*u + (x-1)) by FCONT_1:def 4
           .= (u to_power x) + (-x*u + x-1);
        hence thesis;
  end;
