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
  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
    integral(e(#)f,c,d) = e*integral(f,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 '];
  now
    assume
A2: not c <= d;
    then
A3: integral(f,c,d) = -integral(f,[' d,c ']) by INTEGRA5:def 4;
    thus integral(e(#)f,c,d) = -integral(e(#)f,[' d,c ']) by A2,INTEGRA5:def 4
      .= -integral(e(#)f,d,c) by A2,INTEGRA5:def 4
      .= -e*integral(f,d,c) by A1,A2,Lm12
      .= e*(-integral(f,d,c))
      .= e*integral(f,c,d) by A2,A3,INTEGRA5:def 4;
  end;
  hence thesis by A1,Lm12;
end;
