let n be non empty Element of NAT ; :: thesis: for f being PartFunc of REAL,(REAL n)
for x0 being real number holds
( f is_differentiable_in x0 iff for i being Element of NAT st 1 <= i & i <= n holds
(Proj (i,n)) * f is_differentiable_in x0 )
let f be PartFunc of REAL,(REAL n); :: thesis: for x0 being real number holds
( f is_differentiable_in x0 iff for i being Element of NAT st 1 <= i & i <= n holds
(Proj (i,n)) * f is_differentiable_in x0 )
let x0 be real number ; :: thesis: ( f is_differentiable_in x0 iff for i being Element of NAT st 1 <= i & i <= n holds
(Proj (i,n)) * f is_differentiable_in x0 )
thus ( f is_differentiable_in x0 implies for i being Element of NAT st 1 <= i & i <= n holds
(Proj (i,n)) * f is_differentiable_in x0 ) :: thesis: ( ( for i being Element of NAT st 1 <= i & i <= n holds
(Proj (i,n)) * f is_differentiable_in x0 ) implies f is_differentiable_in x0 )

proof

assume f is_differentiable_in x0 ; :: thesis: for i being Element of NAT st 1 <= i & i <= n holds
(Proj (i,n)) * f is_differentiable_in x0
then ex g being PartFunc of REAL,(REAL-NS n) st
( f = g & g is_differentiable_in x0 ) by Def1;
hence for i being Element of NAT st 1 <= i & i <= n holds
(Proj (i,n)) * f is_differentiable_in x0 by Th25; :: thesis: verum

end;

assume A1: for i being Element of NAT st 1 <= i & i <= n holds
(Proj (i,n)) * f is_differentiable_in x0 ; :: thesis: f is_differentiable_in x0
reconsider g = f as PartFunc of REAL,(REAL-NS n) by REAL_NS1:def 4;
for i being Element of NAT st 1 <= i & i <= n holds
(Proj (i,n)) * g is_differentiable_in x0 by A1;
then g is_differentiable_in x0 by Th25;
hence f is_differentiable_in x0 by Def1; :: thesis: verum