 reserve h,h1 for 0-convergent non-zero Real_Sequence,
         c,c1 for constant Real_Sequence;

theorem :: corresponds to INTEGRA7:12
  for f,F,g,G be PartFunc of REAL,REAL, I be non empty Interval st
    F is_antiderivative_of f,I & G is_antiderivative_of g,I
    holds F+G is_antiderivative_of f+g,I & F-G is_antiderivative_of f-g,I
proof
    let f,F,g,G be PartFunc of REAL,REAL, I be non empty Interval;
    assume that
A1:  F is_antiderivative_of f,I and
A2:  G is_antiderivative_of g,I;

    I c= dom F & I c= dom G by A1,A2,FDIFF_12:def 1; then
    I c= dom F /\ dom G by XBOOLE_1:19; then
A3: I c= dom(F+G) & I c= dom(F-G) by VALUED_1:def 1,12; then
    (F+G)`\I = F`\I + G`\I & (F-G)`\I = F`\I - G`\I by A2,A1,FDIFF_12:19,20;
    hence thesis by A3,A1,A2,RFUNCT_1:44,47,FDIFF_12:19,20;
end;
