let m be non zero Element of NAT ; :: thesis: for i being Element of NAT
for X being Subset of (REAL m)
for f, g being 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 being Element of REAL m st x in X holds
((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) )
let i be Element of NAT ; :: thesis: for X being Subset of (REAL m)
for f, g being 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 being Element of REAL m st x in X holds
((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) )
let X be Subset of (REAL m); :: thesis: for f, g being 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 being Element of REAL m st x in X holds
((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) )
let f, g be PartFunc of (REAL m),REAL; :: thesis: ( X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i implies ( f + g is_partial_differentiable_on X,i & (f + g) `partial| (X,i) = (f `partial| (X,i)) + (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) ) )
assume A1: ( X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i ) ; :: thesis: ( f + g is_partial_differentiable_on X,i & (f + g) `partial| (X,i) = (f `partial| (X,i)) + (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,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 :: thesis: for x being Element of REAL m st x in X holds
( f + g is_partial_differentiable_in x,i & partdiff ((f + g),x,i) = (partdiff (f,x,i)) + (partdiff (g,x,i)) )

let x be Element of REAL m; :: thesis: ( x in X implies ( f + g is_partial_differentiable_in x,i & partdiff ((f + g),x,i) = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) )
assume x in X ; :: thesis: ( f + g is_partial_differentiable_in x,i & partdiff ((f + g),x,i) = (partdiff (f,x,i)) + (partdiff (g,x,i)) )
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; :: thesis: verum

end;

then A6: for x being 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 :: thesis: for x being Element of REAL m st x in X holds
((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i))

let x be Element of REAL m; :: thesis: ( x in X implies ((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) )
assume A10: x in X ; :: thesis: ((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i))
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; :: thesis: verum

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 :: thesis: for x being Element of REAL m st x in X holds
((f + g) `partial| (X,i)) . x = ((f `partial| (X,i)) + (g `partial| (X,i))) . x

let x be Element of REAL m; :: thesis: ( x in X implies ((f + g) `partial| (X,i)) . x = ((f `partial| (X,i)) + (g `partial| (X,i))) . x )
assume A12: x in X ; :: thesis: ((f + g) `partial| (X,i)) . x = ((f `partial| (X,i)) + (g `partial| (X,i))) . 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 ; :: thesis: verum

end;

hence ( f + g is_partial_differentiable_on X,i & (f + g) `partial| (X,i) = (f `partial| (X,i)) + (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) ) by A6, A4, Th60, A1, A8, A9, A11, A3, PARTFUN1:5; :: thesis: verum