
theorem Th35:
  for a, b, c, d being Real st a < b & c <= d for x being
Real st a <= x & x <= b holds L[01](a,b,c,d).x = ((d - c)/(b - a)) * (x
  - a) + c
proof
A1: 0 = (#)(0,1) & 1 = (0,1)(#) by TREAL_1:def 1,def 2;
  let a, b, c, d be Real;
  assume
A2: a < b;
  set G = P[01](a,b,(#)(0,1),(0,1)(#));
  set F = L[01]((#)(c,d),(c,d)(#));
  set f = L[01](a,b,c,d);
  assume
A3: c <= d;
  then
A4: (#)(c,d) = c & (c,d)(#) = d by TREAL_1:def 1,def 2;
  let x be Real;
  assume
A5: a <= x;
  set X = (x-a)/(b-a);
  assume
A6: x <= b;
  then
A7: X in the carrier of Closed-Interval-TSpace (0,1) by A5,Th2;
  x in [.a,b.] by A5,A6,XXREAL_1:1;
  then
A8: x in the carrier of Closed-Interval-TSpace (a,b) by A2,TOPMETR:18;
  then x in dom G by FUNCT_2:def 1;
  then f.x = F.(G.x) by FUNCT_1:13
    .= F.(((b-x)*0 + (x-a)*1)/(b-a)) by A2,A8,A1,TREAL_1:def 4
    .= (1 - X)*c + X * d by A3,A4,A7,TREAL_1:def 3
    .= ((d - c)/(b - a)) * (x - a) + c by XCMPLX_1:234;
  hence thesis;
end;
