let n, m be non zero Nat; for g1, g2 being PartFunc of (REAL m),(REAL n)
for y0 being Element of REAL m st g1 is_differentiable_in y0 & g2 is_differentiable_in y0 holds
( g1 - g2 is_differentiable_in y0 & diff ((g1 - g2),y0) = (diff (g1,y0)) - (diff (g2,y0)) )
let g1, g2 be PartFunc of (REAL m),(REAL n); for y0 being Element of REAL m st g1 is_differentiable_in y0 & g2 is_differentiable_in y0 holds
( g1 - g2 is_differentiable_in y0 & diff ((g1 - g2),y0) = (diff (g1,y0)) - (diff (g2,y0)) )
let y0 be Element of REAL m; ( g1 is_differentiable_in y0 & g2 is_differentiable_in y0 implies ( g1 - g2 is_differentiable_in y0 & diff ((g1 - g2),y0) = (diff (g1,y0)) - (diff (g2,y0)) ) )
assume A1:
( g1 is_differentiable_in y0 & g2 is_differentiable_in y0 )
; ( g1 - g2 is_differentiable_in y0 & diff ((g1 - g2),y0) = (diff (g1,y0)) - (diff (g2,y0)) )
reconsider f1 = g1 as PartFunc of (REAL-NS m),(REAL-NS n) by Th1;
reconsider f2 = g2 as PartFunc of (REAL-NS m),(REAL-NS n) by Th1;
reconsider x0 = y0 as Point of (REAL-NS m) by REAL_NS1:def 4;
( f1 is_differentiable_in x0 & f2 is_differentiable_in x0 )
by A1, Th2;
then A2:
( f1 - f2 is_differentiable_in x0 & diff ((f1 - f2),x0) = (diff (f1,x0)) - (diff (f2,x0)) )
by NDIFF_1:36;
f1 - f2 = g1 - g2
by Th5;
hence
g1 - g2 is_differentiable_in y0
by A2, Th2; diff ((g1 - g2),y0) = (diff (g1,y0)) - (diff (g2,y0))
then A3:
diff ((g1 - g2),y0) = diff ((f1 - f2),x0)
by Th3, Th5;
( diff (f1,x0) = diff (g1,y0) & diff (f2,x0) = diff (g2,y0) )
by Th3, A1;
hence
diff ((g1 - g2),y0) = (diff (g1,y0)) - (diff (g2,y0))
by A2, A3, Th8; verum