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

theorem  :: corresponds to INTEGRA7:14
  for f,g,F,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+F(#)g,I
proof
    let f,g,F,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;
A3: (F(#)G)`\I = G(#)(F`\I) + F(#)(G`\I) by A1,A2,FDIFF_12:24;
    G(#)(F`\I) = (G(#)f)|I & F(#)(G`\I) = (F(#)g)|I by A1,A2,RFUNCT_1:45;
    hence F(#)G is_antiderivative_of f(#)G+F(#)g,I
     by A3,A1,A2,FDIFF_12:24,RFUNCT_1:44;
end;
