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

theorem  :: corresponds to INTEGRA7:15
  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 &
   (for x be set st x in I holds G.x <> 0) holds
    F/G is_antiderivative_of (f(#)G-F(#)g)/(G(#)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 and
A3:  for x be set st x in I holds G.x <> 0;

A4: I c= dom F & I c= dom G by A1,A2,FDIFF_12:def 1;

A5: dom(F/G) = dom F /\ (dom G \ G"{0}) by RFUNCT_1:def 1;

    now let x be set;
     assume A6: x in I; then
     G.x <> 0 by A3; then
     not G.x in {0} by TARSKI:def 1; then
     not x in G"{0} by FUNCT_1:def 7; then
     x in dom G \ G"{0} by A4,A6,XBOOLE_0:def 5;
     hence x in dom(F/G) by A4,A5,A6,XBOOLE_0:def 4;
    end; then
A7: I c= dom(F/G); then
A8:(F/G)`\I = ((F`\I)(#)G - (G`\I)(#)F)/(G^2) by A1,A2,FDIFF_12:25;
    (F`\I)(#)G = (f(#)G)|I & (G`\I)(#)F = (g(#)F)|I by A1,A2,RFUNCT_1:45; then
    (F`\I)(#)G - (G`\I)(#)F = (f(#)G - g(#)F)|I by RFUNCT_1:47;
    hence F/G is_antiderivative_of (f(#)G-F(#)g)/(G(#)G),I
     by A7,A8,A1,A2,FDIFF_12:25,RFUNCT_1:48;
end;
