reserve A for non empty closed_interval Subset of REAL;

theorem
for a,b,c being Real, f,g,h be Function of REAL,REAL st
a <= b & b <= c &
f | ['a,c'] is bounded & g | ['a,c'] is bounded &
h = (f| [.a,b.]) +* (g | [.b,c.]) & f.b = g.b
holds
h | ['a,c'] is bounded
proof
 let a,b,c be Real, f,g,h be Function of REAL,REAL;
 assume that
 A1: a <= b & b <= c and
 A2:  f | ['a,c'] is bounded and
 A3:  g | ['a,c'] is bounded and
 A4: h = (f| [.a,b.]) +* (g | [.b,c.]) and
 A5: f.b = g.b;
 B10: (f| [.a,b.]) tolerates (g | [.b,c.])
 proof
 for x being object st x in (dom (f| [.a,b.])) /\ (dom (g | [.b,c.]))
 holds (f| [.a,b.]) . x = (g | [.b,c.]) . x
 proof
  let x be object;
  assume T1: x in (dom (f| [.a,b.])) /\ (dom (g | [.b,c.]));
  T2: (dom (f| [.a,b.])) /\ (dom (g | [.b,c.]))
   = [.a,b.] /\ (dom (g | [.b,c.])) by FUNCT_2:def 1
  .= [.a,b.] /\ [.b,c.] by FUNCT_2:def 1
  .= {b} by XXREAL_1:418,A1; then
  (f| [.a,b.]) . x = (f| [.a,b.]) . b by TARSKI:def 1,T1
  .= g.b by A5,FUNCT_1:49,XXREAL_1:1,A1
  .= (g | [.b,c.]) . b by FUNCT_1:49,XXREAL_1:1,A1;
  hence thesis by TARSKI:def 1,T1,T2;
 end;
 hence thesis by PARTFUN1:def 4;
 end;
  ex r being Real st for y being set st y in dom (h | ['a,c']) holds
  |.(h | ['a,c']).y.| < r
  proof
   consider r being Real such that
   C1: for y being set st y in dom (f | ['a,c']) holds |.(f | ['a,c']).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,c']) holds |.(g | ['a,c']).y.| < r1
               by COMSEQ_2:def 3,A3;
   take max(r,r1);
   for x being set st x in dom (h | ['a,c']) holds
       |. (h | ['a,c']).x .| < max(r,r1)
   proof
    let x be set;
    assume
    B1: x in dom (h | ['a,c']);
    D1: ['a,c'] = dom (f | ['a,c']) by FUNCT_2:def 1;
    D2: ['a,c'] = dom (g | ['a,c']) by FUNCT_2:def 1;
    B2: x in ['a,c'] by B1;
    reconsider x as Real by B1;
    x in [.a,c.] by INTEGRA5:def 3,A1,XXREAL_0:2,B2; then
    BB2: a <= x & x <= c by XXREAL_1:1;
    per cases;
    suppose S1: x < b; then
     x in [.a,b.] by BB2; then
     B4: x in dom (f| [.a,b.]) by FUNCT_2:def 1;
     E1: |. (h | ['a,c']).x .|
     = |.((f| [.a,b.]) +* (g | [.b,c.])).x.| by A4,FUNCT_1:49,B1
     .= |. (f| [.a,b.]).x .| by FUNCT_4:15,B4,B10
     .= |. (f ).x .| by FUNCT_1:49,XXREAL_1:1,BB2,S1
     .= |. (f | ['a,c']).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 >= b; then
     x in [.b,c.] by BB2; then
     D4:  x in dom (g|[.b,c.]) by FUNCT_2:def 1;
     E2: |. (h | ['a,c']).x .|
     = |.((f| [.a,b.]) +* (g | [.b,c.])).x.| by A4,FUNCT_1:49,B1
     .= |. (g|[.b,c.]).x .| by FUNCT_4:13,D4
     .= |. g.x .| by FUNCT_1:49,BB2,XXREAL_1:1,S2
     .= |. (g | ['a,c']).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;
