reserve a,b,c,d for Real;

theorem Th16:
  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)*1 + (r-a)*0)/(b-a) by A2,A1,Def4
      .= (b-r)/(b-a);
    thus (L*P).c = L.(P.c) by FUNCT_2:15
      .= (1-((b-r)/(b-a)))*b + ((b-r)/(b-a))*a by A2,A4,A5,Def3
      .= ((1*(b-a)-(b-r))/(b-a))*b + ((b-r)/(b-a))*a by A3,XCMPLX_1:127
      .= ((r-a)/(b-a))*(b/1) + ((b-r)/(b-a))*a
      .= ((r-a)*b)/(1*(b-a)) + ((b-r)/(b-a))*a by XCMPLX_1:76
      .= ((r-a)*b)/(b-a) + ((b-r)/(b-a))*(a/1)
      .= ((r-a)*b)/(b-a) + ((b-r)*a)/(1*(b-a)) by XCMPLX_1:76
      .= ((b*r-b*a) + (b-r)*a)/(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)*b + r*a by A2,A4,Def3
      .= r*(a-b) + b;
    thus (P*L).c = P.(L.c) by FUNCT_2:15
      .= ((b-(r*(a-b) + b))*1 + ((r*(a-b) + b)-a)*0)/(b-a) by A2,A1,A6,Def4
      .= (r*(-(a-b)))/(b-a)
      .= c by A3,XCMPLX_1:89;
  end;
  hence thesis by FUNCT_2:124;
end;
