reserve n for non zero Element of NAT;
reserve a,b,r,t for Real;
reserve
  Z for open Subset of REAL,
 y0 for VECTOR of REAL-NS n,
  G for Function of REAL-NS n,REAL-NS n;

theorem Th48:
for f,g be PartFunc of REAL,REAL st a <= t &
    ['a,t'] c= dom f & f is_integrable_on ['a,t'] &
    f| ['a,t'] is bounded &
    ['a,t'] c= dom g & g is_integrable_on ['a,t'] &
    g| ['a,t'] is bounded &
   for x be Real st x in ['a,t'] holds f.x <= g.x holds
  integral(f,a,t) <= integral(g,a,t)
proof
  let f,g be PartFunc of REAL,REAL;
  assume A1: a <= t &
    ['a,t'] c= dom f & f is_integrable_on ['a,t'] &
    f| ['a,t'] is bounded &
    ['a,t'] c= dom g & g is_integrable_on ['a,t'] &
    g| ['a,t'] is bounded &
    for x be Real st x in ['a,t'] holds f.x <= g.x;
  set f0 = f| ['a,t'];
  set g0 = g| ['a,t'];
A2: dom f0 = ['a,t'] by A1,RELAT_1:62;
  rng f0 c= REAL; then
  reconsider f0 as Function of ['a,t'],REAL by A2,FUNCT_2:2;
A3: dom g0 = ['a,t'] by A1,RELAT_1:62;
  rng g0 c= REAL; then
  reconsider g0 as Function of ['a,t'],REAL by A3,FUNCT_2:2;
A4: f0| ['a,t'] is bounded by A1;
A5: f0 is integrable by A1;
A6: g0| ['a,t'] is bounded by A1;
A7: g0 is integrable by A1;
  now let x be Real;
    assume A8: x in ['a,t'];
  A9: f0.x = f.x by A8,FUNCT_1:49;
    g0.x = g.x by A8,FUNCT_1:49;
    hence f0.x <= g0.x by A9,A1,A8;
  end; then
A10: integral(f,['a,t']) <= integral(g,['a,t']) by A4,A5,A6,A7,INTEGRA2:34;
  integral(f,['a,t']) = integral(f,a,t) by A1,INTEGRA5:def 4;
  hence integral(f,a,t) <= integral(g,a,t) by A10,A1,INTEGRA5:def 4;
end;
