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

theorem Th42:
  for f,F be PartFunc of REAL,REAL, I,J be non empty Interval st
   inf I < sup I & I c= J & F is_antiderivative_of f,J holds
    F is_antiderivative_of f,I
proof
    let f,F be PartFunc of REAL,REAL, I,J be non empty Interval;
    assume that
A1:  inf I < sup I and
A2:  I c= J and
A3:  F is_antiderivative_of f,J;

    F is_differentiable_on_interval I by A1,A2,A3,FDIFF_12:38; then
A4: dom(F`\I) = I by FDIFF_12:def 2;

    dom(f|J) = J by A3,FDIFF_12:def 2; then
    dom f /\ J = J by RELAT_1:61; then
A5: dom(f|I) = I by A2,XBOOLE_1:18,RELAT_1:62;
    now let x be Element of REAL;
     assume
A6:   x in dom(F`\I); then
     (F`\I).x = (f|J).x by A4,A3,A1,A2,FDIFF_12:38; then
     (F`\I).x = ((f|J)|I).x by A6,A4,FUNCT_1:49;
     hence (F`\I).x = (f|I).x by A2,RELAT_1:74;
    end;
    hence F is_antiderivative_of f,I by A4,A5,A1,A2,A3,FDIFF_12:38,PARTFUN1:5;
end;
