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 Th10:
  a <= b & [' a,b '] c= X & F is_differentiable_on X &
  F`|X is_integrable_on [' a,b '] & (F`|X)|[' a,b '] is bounded
  implies F.b = integral (F`|X,a,b) + F.a
proof
  assume that
A1: a <= b and
A2: [' a,b '] c= X & F is_differentiable_on X & F`|X is_integrable_on ['
  a,b '] & (F`|X)|[' a,b '] is bounded;
  integral(F`|X,a,b) = integral(F`|X,[' a,b ']) by A1,INTEGRA5:def 4;
  then
A3: integral(F`|X,a,b) = F.(upper_bound [' a,b ']) - F.(lower_bound [' a,b
  ']) by A2,INTEGRA5:13;
A4: [' a,b '] = [.a,b.] by A1,INTEGRA5:def 3;
  then
A5: [.a,b.] = [.lower_bound [.a,b.], upper_bound [.a,b.].] by INTEGRA1:4;
  then a=lower_bound [.a,b.] by A4,INTEGRA1:5;
  hence thesis by A4,A5,A3,INTEGRA1:5;
end;
