reserve A for non empty closed_interval Subset of REAL;

theorem Th18X:
for c being Real, f,g be Function of REAL,REAL st
f | A is bounded & g | A is bounded
holds
( (f | ].-infty,c.[) +* (g | [.c,+infty.[) ) | A is bounded
proof
 let c being Real;
 let f,g be Function of REAL,REAL;
 assume that
 A2: f | A is bounded and
 A4: g | A is bounded;
 set F = ((f|].-infty,c.[) +* (g|[.c,+infty.[));
 reconsider F as Function of REAL,REAL by Th19;
  ex r being Real st for y being set st y in dom (F | A) holds
  |.(F | A).y.| < r
  proof
   consider r being Real such that
   C1: for y being set st y in dom (f | A) holds |.(f | A).y.| < r
               by COMSEQ_2:def 3,A2;
   consider r1 being Real such that
   C2: for y being set st y in dom (g | A) holds |.(g | A).y.| < r1
               by COMSEQ_2:def 3,A4;
   take max(r,r1);
   for x being set st x in dom (F | A) holds |. (F | A).x .| < max(r,r1)
   proof
    let x be set;
    assume B1: x in dom (F | A);
    D1: A = dom (f | A) by FUNCT_2:def 1;
    D2: A = dom (g | A) by FUNCT_2:def 1;
    reconsider x as Real by B1;
    per cases;
    suppose S1: x < c; then
     B4: not ( x in dom (g|[.c,+infty.[)) by XXREAL_1:236;
     E1: |. (F | A).x .| = |. F.x .| by FUNCT_1:49,B1
     .= |. (f|].-infty,c.[).x .| by FUNCT_4:11,B4
     .= |. (f ).x .| by FUNCT_1:49,XXREAL_1:233,S1
     .= |. (f | A).x .| by FUNCT_1:49,B1;
     r <= max(r,r1) by XXREAL_0:25;
     hence thesis by XXREAL_0:2,E1,C1, B1, D1;
    end;
    suppose S2: x >= c; then
     x in [.c,+infty.[ by XXREAL_1:236; then
     D4: x in dom (g|[.c,+infty.[) by FUNCT_2:def 1;
     E2: |. (F | A).x .| = |. F.x .| by FUNCT_1:49,B1
     .= |. (g|[.c,+infty.[).x .| by FUNCT_4:13,D4
     .= |. g.x .| by FUNCT_1:49,XXREAL_1:236,S2
     .= |. (g | A).x .| by FUNCT_1:49,B1;
     r1 <= max(r,r1) by XXREAL_0:25;
     hence thesis by XXREAL_0:2,C2,B1, D2,E2;
    end;
   end;
   hence thesis;
  end;
  hence thesis by COMSEQ_2:def 3;
end;
