reserve a,b,r for Real;
reserve A for non empty set;
reserve X,x for set;
reserve f,g,F,G for PartFunc of REAL,REAL;
reserve n for Element of NAT;

theorem Th18:
  a <= b & [' a,b '] c= X & F is_integral_of f,X &
  f is_integrable_on [' a,b '] & f|[' a,b '] is bounded implies
  F.b = integral(f,a,b) + F.a
proof
  assume that
A1: a <= b and
A2: [' a,b '] c= X;
  assume
A3: F is_integral_of f,X;
  then
A4: F is_differentiable_on X & F`|X=f|X by Lm1;
  assume f is_integrable_on [' a,b '];
  then
A5: f||[' a,b '] is integrable;
  assume
A6: f|[' a,b '] is bounded;
  (F`|X)||[' a,b '] =(f|X )|[' a,b '] by A3,Lm1;
  then
A7: (F`|X)||[' a,b '] = f||[' a,b '] by A2,FUNCT_1:51;
  then F`|X is_integrable_on [' a,b '] by A5;
  then F.b = integral(f|X,a,b) +F.a by A1,A2,A4,A7,A6,Th10;
  then F.b = integral(f|X,[' a,b ']) +F.a by A1,INTEGRA5:def 4;
  then F.b =integral(f,[' a,b ']) +F.a by A2,FUNCT_1:51;
  hence thesis by A1,INTEGRA5:def 4;
end;
