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

theorem Th66:
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;
A3:dom (f`partial|(X,i)) = X & dom (g`partial|(X,i)) = X by Def6,A1;
   dom (f-g) = dom f /\ dom g by VALUED_1:12; 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_1:31;
   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_1:12;
   now let x be Element of REAL m;
    assume A12: x in X;
    hence ((f-g)`partial|(X,i)).x
     = ((f-g)`partial|(X,i))/.x by 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_1:13;
   end;
   hence thesis by A8,A11,A3,A6,A9,A4,Th60,A1,PARTFUN1:5;
end;
