reserve m,n for non zero Element of NAT;
reserve i,j,k for Element of NAT;
reserve Z for set;

theorem Th67:
for X be Subset of REAL m, r be Real, f be PartFunc of REAL m,REAL st
  X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i
holds
   r(#)f is_partial_differentiable_on X,i
 & (r(#)f) `partial|(X,i) = r(#)(f`partial|(X,i))
 & for x be Element of REAL m st x in X holds
    (r(#)f)`partial|(X,i)/.x = r*partdiff(f,x,i)
proof
   let X be Subset of REAL m, r be Real, f be PartFunc of REAL m,REAL;
   assume A1: X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i;
A2:dom (f`partial|(X,i)) = X by Def6,A1;
A3:X c= dom (r(#)f) by A1,VALUED_1:def 5;
A4:
   now let x be Element of REAL m;
    assume x in X; then
    f is_partial_differentiable_in x,i by A1,Th60;
    hence r(#)f is_partial_differentiable_in x,i
       & partdiff(r(#)f,x,i) = r*partdiff(f,x,i) by PDIFF_1:33;
   end; then
A5:for x be Element of REAL m st x in X
     holds r(#)f is_partial_differentiable_in x,i; then
A6:r(#)f is_partial_differentiable_on X,i by A3,Th60,A1; then
A7:dom ((r(#)f)`partial|(X,i)) = X by Def6;
A8:
   now let x be Element of REAL m;
    assume A9:x in X; then
    ((r(#)f)`partial|(X,i))/.x = partdiff((r(#)f),x,i) by A6,Def6;
    hence ((r(#)f)`partial|(X,i))/.x = r*partdiff(f,x,i) by A4,A9;
   end;
   dom (r(#)(f`partial|(X,i))) = dom (f`partial|(X,i)) by VALUED_1:def 5; then
A10:dom (r(#)(f`partial|(X,i))) = X by Def6,A1;
   now let x be Element of REAL m;
    assume A11: x in X;
    hence ((r(#)f)`partial|(X,i)).x
      = ((r(#)f)`partial|(X,i))/.x by A7,PARTFUN1:def 6
     .= r*partdiff(f,x,i) by A8,A11
     .= r*((f`partial|(X,i))/.x) by A11,Def6,A1
     .= r*((f`partial|(X,i)).x) by A11,A2,PARTFUN1:def 6
     .= (r(#)(f`partial|(X,i))).x by A11,A10,VALUED_1:def 5;
   end;
   hence thesis by A7,A10,A5,A8,A3,Th60,A1,PARTFUN1:5;
end;
