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

theorem Th17:
  for f be PartFunc of REAL,REAL, I,J be non empty Interval st
   f is_differentiable_on_interval I & J c= I & inf J < sup J holds
    (f`\I)|J = (f`\J)
proof
    let f be PartFunc of REAL,REAL, I,J be non empty Interval;
    assume that
A1:  f is_differentiable_on_interval I and
A2:  J c= I and
A3:  inf J < sup J;
    f is_differentiable_on_interval J by A1,A2,A3,FDIFF_12:38; then
A4: dom(f`\I) = I & dom(f`\J) = J by A1,FDIFF_12:def 2;
    for x be Element of REAL st x in dom((f`\I)|J) holds
     ((f`\I)|J).x = (f`\J).x
    proof
     let x be Element of REAL;
     assume A5: x in dom((f`\I)|J); then
     ((f`\I)|J).x = (f`\I).x by FUNCT_1:47;
     hence ((f`\I)|J).x = (f`\J).x by A1,A2,A3,A5,FDIFF_12:38;
    end;
    hence (f`\I)|J = (f`\J) by A4,A2,RELAT_1:62,PARTFUN1:5;
end;
