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

theorem Th62:
for X be Subset of REAL m,
    f be PartFunc of REAL m,REAL, g be PartFunc of REAL m,REAL 1, i being Nat
   st
<>*f = g & X is open & 1<=i & i<=m & f is_partial_differentiable_on X,i holds
  f`partial|(X,i) is_continuous_on X iff g`partial|(X,i) is_continuous_on X
proof
   let X be Subset of REAL m;
   let f be PartFunc of REAL m,REAL, g be PartFunc of REAL m,REAL 1,
       i be Nat;
   assume A1: <>*f = g & X is open & 1 <= i & i <= m
            & f is_partial_differentiable_on X,i; then
A2:g is_partial_differentiable_on X,i by Th61;
   set ff= f`partial|(X,i);
   set gg= g`partial|(X,i);
A3:
   for x,y be Element of REAL m st x in X & y in X holds
     |. (f`partial|(X,i))/.x - (f`partial|(X,i))/.y .|
       = |. (g`partial|(X,i))/.x - (g`partial|(X,i))/.y .|
   proof
    let x,y be Element of REAL m;
    assume A4: x in X & y in X; then
A5: (f`partial|(X,i))/.x = partdiff(f,x,i) &
    (f`partial|(X,i))/.y = partdiff(f,y,i) by A1,Def6;
A6:(g`partial|(X,i))/.x = partdiff(g,x,i) &
    (g`partial|(X,i))/.y = partdiff(g,y,i) by A2,A4,PDIFF_7:def 5;
    g is_partial_differentiable_in x,i &
    g is_partial_differentiable_in y,i by A2,A4,A1,PDIFF_7:34; then
    partdiff(g,x,i) = <*partdiff(f,x,i)*> &
    partdiff(g,y,i) = <*partdiff(f,y,i)*> by A1,PDIFF_1:19; then
    (g`partial|(X,i))/.x - (g`partial|(X,i))/.y
     = <* (f`partial|(X,i))/.x - (f`partial|(X,i))/.y *> by A5,A6,RVSUM_1:29;
    hence thesis by Lm1;
   end;
A7:dom gg = X by A2,PDIFF_7:def 5;
A8:dom ff = X by Def6,A1;
   hereby assume A9: f`partial|(X,i) is_continuous_on X;
    now let x0 be Element of REAL m, r be Real;
     assume A10: x0 in X & 0<r; then
     consider s be Real such that
A11:  0<s & for x1 be Element of REAL m st x1 in X & |. x1- x0 .| < s
        holds |. ff/.x1 - ff/.x0 .| < r by A8,A9,Th45;
     take s;
     thus 0<s by A11;
     let x1 be Element of REAL m;
     assume A12: x1 in X & |. x1- x0 .| < s; then
     |. ff/.x1 - ff/.x0 .| < r by A11;
     hence |. gg/.x1 - gg/.x0 .| < r by A10,A12,A3;
    end;
    hence g`partial|(X,i) is_continuous_on X by A7,PDIFF_7:38;
   end;
   hereby assume A13: g`partial|(X,i) is_continuous_on X;
    now let x0 be Element of REAL m, r be Real;
     assume A14: x0 in X & 0<r; then
     consider s be Real such that
A15:  0<s & for x1 be Element of REAL m st x1 in X & |. x1- x0 .| < s
       holds |. gg/.x1 - gg/.x0 .| < r by A13,PDIFF_7:38;
     take s;
     thus 0<s by A15;
     let x1 be Element of REAL m;
     assume A16: x1 in X & |. x1- x0 .| < s; then
     |. gg/.x1 - gg/.x0 .| < r by A15;
     hence |. ff/.x1 - ff/.x0 .| < r by A14,A16,A3;
    end;
    hence f`partial|(X,i) is_continuous_on X by Th45,A8;
   end;
end;
