reserve a,b,c,d,e,x,r for Real,
  A for non empty closed_interval Subset of REAL,
  f,g for PartFunc of REAL,REAL;

theorem Th21:
  a <= b & f is_integrable_on [' a,b '] & f|[' a,b '] is bounded &
[' a,b '] c= dom f & c in [' a,b '] & d in [' a,b '] implies [' min(c,d),max(c,
d) '] c= dom (abs f) & abs f is_integrable_on [' min(c,d),max(c,d) '] & abs f|
[' min(c,d),max(c,d) '] is bounded & |.integral(f,c,d).| <= integral(abs f,min(
  c,d),max(c,d))
proof
  assume
A1: a <= b & f is_integrable_on [' a,b '] & f|[' a,b '] is bounded & ['
  a,b '] c= dom f & c in [' a,b '] & d in [' a,b '];
A2: now
    assume
A3: not c <= d;
    then integral(f,c,d) = -integral(f,[' d,c ']) by INTEGRA5:def 4;
    then integral(f,c,d) = -integral(f,d,c) by A3,INTEGRA5:def 4;
    then
A4: |.integral(f,c,d).| = |.integral(f,d,c).| by COMPLEX1:52;
    d = min(c,d) & c = max(c,d) by A3,XXREAL_0:def 9,def 10;
    hence thesis by A1,A3,A4,Lm5;
  end;
  now
    assume
A5: c <= d;
    then c = min(c,d) & d = max(c,d) by XXREAL_0:def 9,def 10;
    hence thesis by A1,A5,Lm5;
  end;
  hence thesis by A2;
end;
