reserve a,b,r for Real;
reserve A for non empty set;
reserve X,x for set;
reserve f,g,F,G for PartFunc of REAL,REAL;
reserve n for Element of NAT;

theorem
  F is_integral_of f,X implies r(#)F is_integral_of r(#)f,X
proof
  assume
A1: F is_integral_of f,X;
  then
A2: F is_differentiable_on X by Lm1;
  then
A3: r(#)F is_differentiable_on X by Th7;
A4: F`|X = f|X by A1,Lm1;
  then dom(f|X) = X by A2,FDIFF_1:def 7;
  then dom f /\ X = X by RELAT_1:61;
  then X c= dom f by XBOOLE_1:18;
  then
A5: X c= dom(r(#)f) by VALUED_1:def 5;
A6: now
    reconsider r1 = r as Real;
    let x be Element of REAL;
    assume x in dom((r(#)F)`|X);
    then
A7: x in X by A3,FDIFF_1:def 7;
    then F|X is_differentiable_in x by A2;
    then
A8: F is_differentiable_in x by A7,Th4;
    ((r(#)F)`|X).x = diff(r(#)F,x) by A3,A7,FDIFF_1:def 7;
    then
A9: ((r1(#)F)`|X).x = r1*diff(F,x) by A8,FDIFF_1:15;
    (F`|X).x = diff(F,x) by A2,A7,FDIFF_1:def 7;
    then f.x = diff(F,x) by A4,A7,FUNCT_1:49;
    then ((r(#)F)`|X).x = (r(#)f).x by A5,A7,A9,VALUED_1:def 5;
    hence ((r(#)F)`|X).x = ((r(#)f)|X).x by A7,FUNCT_1:49;
  end;
  X = dom((r(#)f)|X) by A5,RELAT_1:62;
  then dom((r(#)F)`|X) = dom((r(#)f)|X) by A3,FDIFF_1:def 7;
  then (r(#)F)`|X = (r(#)f)|X by A6,PARTFUN1:5;
  hence thesis by A3,Lm1;
end;
