reserve a,b,c,d for Real;

theorem Th11:
  a < b implies for t1,t2 being Point of Closed-Interval-TSpace(0,1)
  for s being Point of Closed-Interval-TSpace(a,b) holds
  P[01](a,b,t1,t2).s = ((t2 - t1)/(b-a))*s + (b*t1 -a*t2)/(b-a)
proof
  assume
A1: a < b;
  let t1,t2 be Point of Closed-Interval-TSpace(0,1);
  let s be Point of Closed-Interval-TSpace(a,b);
  thus P[01](a,b,t1,t2).s = ((b-s)*t1 + (s-a)*t2)/(b-a) by A1,Def4
    .= (s*(t2 - t1) + (b*t1 -a*t2))/(b-a)
    .= (s*(t2 - t1))/(b-a) + (b*t1 -a*t2)/(b-a) by XCMPLX_1:62
    .= (s*(t2 - t1))* (1/(b-a)) + (b*t1 -a*t2)/(b-a) by XCMPLX_1:99
    .= ((t2 - t1)* (1/(b-a)))*s + (b*t1 -a*t2)/(b-a)
    .= ((t2 - t1)/(b-a))*s + (b*t1 -a*t2)/(b-a) by XCMPLX_1:99;
end;
