reserve a,b,c,d for Real;

theorem
  a < b implies for t1,t2 being Point of Closed-Interval-TSpace(0,1)
  holds P[01](a,b,t1,t2).(#)(a,b) = t1 & P[01](a,b,t1,t2).(a,b)(#) = t2
proof
  assume
A1: a < b;
  then
A2: b - a <> 0;
  let t1,t2 be Point of Closed-Interval-TSpace(0,1);
  reconsider r1 = t1, r2 = t2 as Real;
  a = (#)(a,b) by A1,Def1;
  hence P[01](a,b,t1,t2).(#)(a,b) = ((b-a)*r1 + (a-a)*r2)/(b-a) by A1,Def4
    .= t1 by A2,XCMPLX_1:89;
  b = (a,b)(#) by A1,Def2;
  hence P[01](a,b,t1,t2).(a,b)(#) = ((b-b)*r1 + (b-a)*r2)/(b-a) by A1,Def4
    .= t2 by A2,XCMPLX_1:89;
end;
