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

theorem Th65:
for X be Subset of REAL m, f,g be PartFunc of REAL m,REAL st
 X is open & 1 <= i & i <= m
  & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i
 holds
   (f+g) is_partial_differentiable_on X,i
 & (f+g)`partial|(X,i) = f`partial|(X,i) + g`partial|(X,i)
 & for x be Element of REAL m st x in X holds
     (f+g)`partial|(X,i)/.x = partdiff(f,x,i) + partdiff(g,x,i)
proof
   let X be Subset of REAL m, f,g be PartFunc of REAL m,REAL;
   assume A1: X is open & 1 <= i & i <= m
   & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i;
     then
A3:dom(f`partial|(X,i)) = X & dom(g`partial|(X,i)) = X by Def6;
   dom (f+g) = dom f /\ dom g by VALUED_1:def 1; then
A4:X c= dom(f+g) by A1,XBOOLE_1:19;
A5:
   now let x be Element of REAL m;
    assume x in X; then
    f is_partial_differentiable_in x,i &
    g is_partial_differentiable_in x,i by A1,Th60;
    hence f+g is_partial_differentiable_in x,i
     & partdiff(f+g,x,i) = partdiff(f,x,i) + partdiff(g,x,i) by PDIFF_1:29;
   end; then
A6:for x be Element of REAL m st x in X
     holds f+g is_partial_differentiable_in x,i; then
A7:(f+g) is_partial_differentiable_on X,i by A4,Th60,A1; then
A8:dom ((f+g)`partial|(X,i)) = X by Def6;
A9:now let x be Element of REAL m;
    assume A10:x in X; then
    ((f+g)`partial|(X,i))/.x = partdiff(f+g,x,i) by A7,Def6;
    hence ((f+g)`partial|(X,i))/.x = partdiff(f,x,i) + partdiff(g,x,i)
      by A5,A10;
   end;
A11:
   dom ((f`partial|(X,i)) + (g`partial|(X,i)))
     = dom (f`partial|(X,i)) /\ dom (g`partial|(X,i)) by VALUED_1:def 1;
   now let x be Element of REAL m;
    assume A12: x in X;
    thus ((f+g)`partial|(X,i)).x
     = ((f+g)`partial|(X,i))/.x by A12,A8,PARTFUN1:def 6
    .= partdiff(f,x,i) + partdiff(g,x,i) by A9,A12
    .= ((f`partial|(X,i))/.x) + partdiff(g,x,i) by A12,Def6,A1
    .= ((f`partial|(X,i))/.x) + ((g`partial|(X,i))/.x) by A12,Def6,A1
    .= ((f`partial|(X,i)).x) + ((g`partial|(X,i))/.x) by A12,A3,PARTFUN1:def 6
    .= ((f`partial|(X,i)).x) + ((g`partial|(X,i)).x) by A12,A3,PARTFUN1:def 6
    .= ((f`partial|(X,i))+(g`partial|(X,i))).x by A12,A11,A3,VALUED_1:def 1;
   end;
   hence thesis by A6,A4,Th60,A1,A8,A9,A11,A3,PARTFUN1:5;
end;
