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 Th10:
  a<=b & [' a,b '] c= dom f & f is_integrable_on [' a,b '] & f|['
  a,b '] is bounded implies integral(c(#)f,a,b) =c*integral(f,a,b)
proof
  assume that
A1: a<=b and
A2: [' a,b '] c= dom f & f is_integrable_on [' a,b '] & f|[' a,b '] is
  bounded;
  integral(f,a,b) = integral(f,[' a,b ']) & integral(c(#)f,a,b) = integral
  (c (#)f,[' a,b ']) by A1,INTEGRA5:def 4;
  hence thesis by A2,Th9;
end;
