
theorem Th33:
  for a, b, c, d being Real st a < b & c < d holds L[01](a,
  b,c,d).a = c & L[01](a,b,c,d).b = d
proof
  let a, b, c, d be Real;
  assume that
A1: a < b and
A2: c < d;
  a in [.a,b.] by A1,XXREAL_1:1;
  then a in the carrier of Closed-Interval-TSpace (a,b) by A1,TOPMETR:18;
  then a in dom P[01](a,b,(#)(0,1),(0,1)(#)) by FUNCT_2:def 1;
  hence L[01](a,b,c,d).a = L[01]((#)(c,d),(c,d)(#)).(P[01](a,b,(#)(0,1),(0,1)
  (#)).a) by FUNCT_1:13
    .= L[01]((#)(c,d),(c,d)(#)).(P[01](a,b,(#)(0,1),(0,1)(#)).(#)(a,b)) by A1,
TREAL_1:def 1
    .= L[01]((#)(c,d),(c,d)(#)).((#)(0,1)) by A1,TREAL_1:13
    .= (#)(c,d) by A2,TREAL_1:9
    .= c by A2,TREAL_1:def 1;
  b in [.a,b.] by A1,XXREAL_1:1;
  then b in the carrier of Closed-Interval-TSpace (a,b) by A1,TOPMETR:18;
  then b in dom P[01](a,b,(#)(0,1),(0,1)(#)) by FUNCT_2:def 1;
  hence L[01](a,b,c,d).b = L[01]((#)(c,d),(c,d)(#)).(P[01](a,b,(#)(0,1),(0,1)
  (#)).b) by FUNCT_1:13
    .= L[01]((#)(c,d),(c,d)(#)).(P[01](a,b,(#)(0,1),(0,1)(#)).(a,b)(#)) by A1,
TREAL_1:def 2
    .= L[01]((#)(c,d),(c,d)(#)).((0,1)(#)) by A1,TREAL_1:13
    .= (c,d)(#) by A2,TREAL_1:9
    .= d by A2,TREAL_1:def 2;
end;
