reserve a,b,c,d for Real;

theorem Th15:
  a < b implies
  id Closed-Interval-TSpace(a,b) =
     L[01]((#)(a,b),(a,b)(#)) * P[01](a,b,(#)(0,1),(0,1)(#)) &
  id Closed-Interval-TSpace(0,1) =
     P[01](a,b,(#)(0,1),(0,1)(#)) * L[01]((#)(a,b),(a,b)(#))
proof
A1: 0 = (#)(0,1) & 1 = (0,1)(#) by Def1,Def2;
  set L = L[01]((#)(a,b),(a,b)(#)), P = P[01](a,b,(#)(0,1),(0,1)(#));
  assume
A2: a < b;
  then
A3: b - a <> 0;
A4: a = (#)(a,b) & b = (a,b)(#) by A2,Def1,Def2;
  for c being Point of Closed-Interval-TSpace(a,b) holds (L*P).c = c
  proof
    let c be Point of Closed-Interval-TSpace(a,b);
    reconsider r = c as Real;
A5: P.c = ((b-r)*0 + (r-a)*1)/(b-a) by A2,A1,Def4
      .= (r-a)/(b-a);
    thus (L*P).c = L.(P.c) by FUNCT_2:15
      .= (1-((r-a)/(b-a)))*a + ((r-a)/(b-a))*b by A2,A4,A5,Def3
      .= ((1*(b-a)-(r-a))/(b-a))*a + ((r-a)/(b-a))*b by A3,XCMPLX_1:127
      .= ((b-r)/(b-a))*(a/1) + ((r-a)/(b-a))*b
      .= ((b-r)*a)/(1*(b-a)) + ((r-a)/(b-a))*b by XCMPLX_1:76
      .= ((b-r)*a)/(b-a) + ((r-a)/(b-a))*(b/1)
      .= ((b-r)*a)/(b-a) + ((r-a)*b)/(1*(b-a)) by XCMPLX_1:76
      .= ((a*b-a*r) + (r-a)*b)/(b-a) by XCMPLX_1:62
      .= ((b-a)*r)/(b-a)
      .= c by A3,XCMPLX_1:89;
  end;
  hence id Closed-Interval-TSpace(a,b) = L*P by FUNCT_2:124;
  for c being Point of Closed-Interval-TSpace(0,1) holds (P*L).c = c
  proof
    let c be Point of Closed-Interval-TSpace(0,1);
    reconsider r = c as Real;
A6: L.c = (1-r)*a + r*b by A2,A4,Def3
      .= r*(b-a) + a;
    thus (P*L).c = P.(L.c) by FUNCT_2:15
      .= ((b-(r*(b-a) + a))*0 + ((r*(b-a) + a)-a)*1)/(b-a) by A2,A1,A6,Def4
      .= c by A3,XCMPLX_1:89;
  end;
  hence thesis by FUNCT_2:124;
end;
