:: PDIFF_7 semantic presentation

begin


registration
let n be Nat;
let p, q be Element of (TOP-REAL n);
let f, g be real-valued FinSequence;
identifyp + q with f + g when p = f, q = g;
compatibility
 ( p = f & q = g implies p + q = f + g ) ;
end;

registration
let r, s be real number ;
let n be Nat;
let p be Element of (TOP-REAL n);
let f be real-valued FinSequence;
identifyr * p with s * f when r = s, p = f;
compatibility
 ( r = s & p = f implies r * p = s * f ) ;
end;

registration
let n be Nat;
let p be Element of (TOP-REAL n);
let f be real-valued FinSequence;
identify - p with  - f when p = f;
compatibility
 ( p = f implies - p = - f ) ;
end;

registration
let n be Nat;
let p, q be Element of (TOP-REAL n);
let f, g be real-valued FinSequence;
identifyp - q with f - g when p = f, q = g;
compatibility
 ( p = f & q = g implies p - q = f - g ) ;
end;

          Lm1:  for S being RealNormSpace
for x being Point of S
for N1, N2 being Neighbourhood of x holds N1 /\ N2 is Neighbourhood of x
proof
let S be RealNormSpace; ::_thesis: for x being Point of S
for N1, N2 being Neighbourhood of x holds N1 /\ N2 is Neighbourhood of x
let x be Point of S; ::_thesis: for N1, N2 being Neighbourhood of x holds N1 /\ N2 is Neighbourhood of x
let N1, N2 be Neighbourhood of x; ::_thesis: N1 /\ N2 is Neighbourhood of x
consider N being Neighbourhood of x such that
A1: ( N c= N1 & N c= N2 ) by NDIFF_1:1;
A2: N c= N1 /\ N2 by A1, XBOOLE_1:19;
consider g being Real such that
A3: 0 < g and
A4:  {  y where y is Point of S : ||.(y - x).|| < g  }  c= N by NFCONT_1:def_1;
  {  y where y is Point of S : ||.(y - x).|| < g  }  c= N1 /\ N2 by A2, A4, XBOOLE_1:1;
hence  N1 /\ N2 is Neighbourhood of x by A3, NFCONT_1:def_1; ::_thesis: verum
end;

Lm2:  for i, j being Element of NAT st i <= j holds
i + (j -' i) = (i + j) -' i
proof
let i, j be Element of NAT ; ::_thesis: ( i <= j implies i + (j -' i) = (i + j) -' i )
assume  i <= j ; ::_thesis: i + (j -' i) = (i + j) -' i
then  0 <= j - i by XREAL_1:48;
then A1: j -' i = j - i by XREAL_0:def_2;
 (i + j) -' i = (i + j) - i by XREAL_0:def_2;
hence  i + (j -' i) = (i + j) -' i by A1; ::_thesis: verum
end;

theorem Th1: :: PDIFF_7:1
for i, j being Element of NAT st i <= j holds
(0* j) | i = 0* i
proof
let i, j be Element of NAT ; ::_thesis: ( i <= j implies (0* j) | i = 0* i )
assume A1: i <= j ; ::_thesis: (0* j) | i = 0* i
A2: ((0* i) ^ (0* (j -' i))) | (len (0* i)) = 0* i by FINSEQ_5:23;
 i + (j -' i) = (i + j) -' i by A1, Lm2;
then  i + (j -' i) = (i + j) - i by XREAL_0:def_2;
then  (0* i) ^ (0* (j -' i)) = 0* j by FINSEQ_2:123;
hence  (0* j) | i = 0* i by A2, CARD_1:def_7; ::_thesis: verum
end;

theorem Th2: :: PDIFF_7:2
for i, j being Element of NAT st i <= j holds
(0* j) | (i -' 1) = 0* (i -' 1)
proof
let i, j be Element of NAT ; ::_thesis: ( i <= j implies (0* j) | (i -' 1) = 0* (i -' 1) )
assume  i <= j ; ::_thesis: (0* j) | (i -' 1) = 0* (i -' 1)
then A1: i - 1 <= j - 1 by XREAL_1:9;
 j - 1 <= j by XREAL_1:43;
then  i - 1 <= j by A1, XXREAL_0:2;
then  i -' 1 <= j by XREAL_0:def_2;
hence  (0* j) | (i -' 1) = 0* (i -' 1) by Th1; ::_thesis: verum
end;

Lm3:  for i, j being Element of NAT st i <= j holds
(0* j) /^ i = 0* (j -' i)
proof
let i, j be Element of NAT ; ::_thesis: ( i <= j implies (0* j) /^ i = 0* (j -' i) )
assume A1: i <= j ; ::_thesis: (0* j) /^ i = 0* (j -' i)
 len ((0* j) /^ i) = (len (0* j)) -' i by RFINSEQ:29;
then A2: len ((0* j) /^ i) = j -' i by CARD_1:def_7;
A3: len (0* (j -' i)) = j -' i by CARD_1:def_7;
A4: i <= len (0* j) by A1, CARD_1:def_7;
 for k being Nat st 1 <= k & k <= len ((0* j) /^ i) holds
((0* j) /^ i) . k = (0* (j -' i)) . k
proof
let k be Nat; ::_thesis: ( 1 <= k & k <= len ((0* j) /^ i) implies ((0* j) /^ i) . k = (0* (j -' i)) . k )
assume A5: ( 1 <= k & k <= len ((0* j) /^ i) ) ; ::_thesis: ((0* j) /^ i) . k = (0* (j -' i)) . k
 k in dom ((0* j) /^ i) by A5, FINSEQ_3:25;
then  ((0* j) /^ i) . k = (0* j) . (k + i) by A4, RFINSEQ:def_1;
then A7: ((0* j) /^ i) . k = 0 ;
thus  ((0* j) /^ i) . k = (0* (j -' i)) . k by A7; ::_thesis: verum
end;
hence  (0* j) /^ i = 0* (j -' i) by A2, A3, FINSEQ_1:14; ::_thesis: verum
end;

Lm4:  for i, j being Element of NAT st i > j holds
(0* j) /^ i = 0* (j -' i)
proof
let i, j be Element of NAT ; ::_thesis: ( i > j implies (0* j) /^ i = 0* (j -' i) )
assume A1: i > j ; ::_thesis: (0* j) /^ i = 0* (j -' i)
then A2: i > len (0* j) by CARD_1:def_7;
 0 > j - i by A1, XREAL_1:49;
then A3: j -' i = 0 by XREAL_0:def_2;
 (0* j) /^ i = 0 by A2, RFINSEQ:def_1;
hence  (0* j) /^ i = 0* (j -' i) by A3; ::_thesis: verum
end;

theorem Th3: :: PDIFF_7:3
for j, i being Element of NAT holds (0* j) /^ i = 0* (j -' i)
proof
let j, i be Element of NAT ; ::_thesis: (0* j) /^ i = 0* (j -' i)
percases ( i <= j or i > j ) ;
suppose i <= j ; ::_thesis: (0* j) /^ i = 0* (j -' i)
hence  (0* j) /^ i = 0* (j -' i) by Lm3; ::_thesis: verum
end;
suppose i > j ; ::_thesis: (0* j) /^ i = 0* (j -' i)
hence  (0* j) /^ i = 0* (j -' i) by Lm4; ::_thesis: verum
end;
end;
end;

theorem :: PDIFF_7:4
for i, j being Element of NAT holds
( ( i <= j implies (0* j) | (i -' 1) = 0* (i -' 1) ) & (0* j) /^ i = 0* (j -' i) ) by Th2, Th3;

theorem Th5: :: PDIFF_7:5
for i, j being Element of NAT
for xi being Element of (REAL-NS 1) st 1 <= i & i <= j holds
||.((reproj (i,(0. (REAL-NS j)))) . xi).|| = ||.xi.||
proof
let i, j be Element of NAT ; ::_thesis: for xi being Element of (REAL-NS 1) st 1 <= i & i <= j holds
||.((reproj (i,(0. (REAL-NS j)))) . xi).|| = ||.xi.||
let xi be Element of (REAL-NS 1); ::_thesis: ( 1 <= i & i <= j implies ||.((reproj (i,(0. (REAL-NS j)))) . xi).|| = ||.xi.|| )
assume A1: ( 1 <= i & i <= j ) ; ::_thesis: ||.((reproj (i,(0. (REAL-NS j)))) . xi).|| = ||.xi.||
consider q being Element of REAL , y being Element of REAL j such that
A2: ( xi = <*q*> & y = 0. (REAL-NS j) & (reproj (i,(0. (REAL-NS j)))) . xi = (reproj (i,y)) . q ) by PDIFF_1:def_6;
A3: (reproj (i,(0. (REAL-NS j)))) . xi = Replace (y,i,q) by A2, PDIFF_1:def_5;
 len y = j by CARD_1:def_7;
then  (reproj (i,(0. (REAL-NS j)))) . xi = ((y | (i -' 1)) ^ <*q*>) ^ (y /^ i) by A1, A3, FINSEQ_7:def_1;
then A4: ||.((reproj (i,(0. (REAL-NS j)))) . xi).|| = |.(((y | (i -' 1)) ^ <*q*>) ^ (y /^ i)).| by A2, REAL_NS1:1;
 y | (i -' 1) = (0* j) | (i -' 1) by A2, REAL_NS1:def_4;
then  sqrt (Sum (sqr (y | (i -' 1)))) = |.(0* (i -' 1)).| by A1, Th2;
then  sqrt (Sum (sqr (y | (i -' 1)))) = 0 by EUCLID:7;
then A5: Sum (sqr (y | (i -' 1))) = 0 by RVSUM_1:86, SQUARE_1:24;
 y /^ i = (0* j) /^ i by A2, REAL_NS1:def_4;
then  sqrt (Sum (sqr (y /^ i))) = |.(0* (j -' i)).| by Th3;
then A6: sqrt (Sum (sqr (y /^ i))) = 0 by EUCLID:7;
sqr (((y | (i -' 1)) ^ <*q*>) ^ (y /^ i)) = (sqr ((y | (i -' 1)) ^ <*q*>)) ^ (sqr (y /^ i)) by RVSUM_1:144
.= ((sqr (y | (i -' 1))) ^ (sqr <*q*>)) ^ (sqr (y /^ i)) by RVSUM_1:144
.= ((sqr (y | (i -' 1))) ^ <*(q ^2)*>) ^ (sqr (y /^ i)) by RVSUM_1:55 ;
then  Sum (sqr (((y | (i -' 1)) ^ <*q*>) ^ (y /^ i))) = (Sum ((sqr (y | (i -' 1))) ^ <*(q ^2)*>)) + (Sum (sqr (y /^ i))) by RVSUM_1:75
.= ((Sum (sqr (y | (i -' 1)))) + (q ^2)) + (Sum (sqr (y /^ i))) by RVSUM_1:74
.= q ^2 by A5, A6, RVSUM_1:86, SQUARE_1:24 ;
then A7: ||.((reproj (i,(0. (REAL-NS j)))) . xi).|| = |.q.| by A4, COMPLEX1:72;
 (proj (1,1)) . <*q*> = q by PDIFF_1:1;
hence  ||.((reproj (i,(0. (REAL-NS j)))) . xi).|| = ||.xi.|| by A7, A2, PDIFF_1:4; ::_thesis: verum
end;

theorem Th6: :: PDIFF_7:6
for m, i being Element of NAT
for x being Element of REAL m
for r being Real holds
( ((reproj (i,x)) . r) - x = (reproj (i,(0* m))) . (r - ((proj (i,m)) . x)) & x - ((reproj (i,x)) . r) = (reproj (i,(0* m))) . (((proj (i,m)) . x) - r) )
proof
let m, i be Element of NAT ; ::_thesis: for x being Element of REAL m
for r being Real holds
( ((reproj (i,x)) . r) - x = (reproj (i,(0* m))) . (r - ((proj (i,m)) . x)) & x - ((reproj (i,x)) . r) = (reproj (i,(0* m))) . (((proj (i,m)) . x) - r) )
let x be Element of REAL m; ::_thesis: for r being Real holds
( ((reproj (i,x)) . r) - x = (reproj (i,(0* m))) . (r - ((proj (i,m)) . x)) & x - ((reproj (i,x)) . r) = (reproj (i,(0* m))) . (((proj (i,m)) . x) - r) )
let r be Real; ::_thesis: ( ((reproj (i,x)) . r) - x = (reproj (i,(0* m))) . (r - ((proj (i,m)) . x)) & x - ((reproj (i,x)) . r) = (reproj (i,(0* m))) . (((proj (i,m)) . x) - r) )
reconsider p = ((reproj (i,x)) . r) - x as m -element FinSequence ;
reconsider q = (reproj (i,(0* m))) . (r - ((proj (i,m)) . x)) as m -element FinSequence ;
reconsider s = x - ((reproj (i,x)) . r) as m -element FinSequence ;
reconsider t = (reproj (i,(0* m))) . (((proj (i,m)) . x) - r) as m -element FinSequence ;
A1: ( dom p = Seg m & dom q = Seg m & dom s = Seg m & dom t = Seg m & dom x = Seg m & dom (0* m) = Seg m ) by FINSEQ_1:89;
reconsider x1 = x as Element of m -tuples_on REAL ;
A2: (reproj (i,x)) . r = Replace (x,i,r) by PDIFF_1:def_5;
reconsider y1 = (reproj (i,x)) . r as Element of m -tuples_on REAL ;
A3: ( len x = m & len (0* m) = m ) by A1, FINSEQ_1:def_3;
then A4: len (Replace (x,i,r)) = m by FINSEQ_7:5;
 for k being Nat st k in dom p holds
p . k = q . k
proof
let k be Nat; ::_thesis: ( k in dom p implies p . k = q . k )
assume A5: k in dom p ; ::_thesis: p . k = q . k
then A6: ( 1 <= k & k <= m ) by A1, FINSEQ_1:1;
then  k in dom (Replace (x,i,r)) by A4, FINSEQ_3:25;
then A7: (Replace (x,i,r)) /. k = (Replace (x,i,r)) . k by PARTFUN1:def_6;
A8: p . k = (y1 . k) - (x1 . k) by RVSUM_1:27;
 q = Replace ((0* m),i,(r - ((proj (i,m)) . x))) by PDIFF_1:def_5;
then A9: q . k = (Replace ((0* m),i,(r - ((proj (i,m)) . x)))) /. k by A5, A1, PARTFUN1:def_6;
percases ( k = i or k <> i ) ;
supposeA10: k = i ; ::_thesis: p . k = q . k
then  ( p . k = r - (x1 . k) & q . k = r - ((proj (i,m)) . x) ) by A2, A3, A6, A7, A8, A9, FINSEQ_7:8;
hence  p . k = q . k by A10, PDIFF_1:def_1; ::_thesis: verum
end;
suppose k <> i ; ::_thesis: p . k = q . k
then  ( (Replace (x,i,r)) . k = x1 /. k & q . k = (0* m) /. k ) by A3, A6, A7, A9, FINSEQ_7:10;
then  ( (Replace (x,i,r)) . k = x1 . k & q . k = (m |-> 0) . k ) by A5, A1, PARTFUN1:def_6;
hence  p . k = q . k by A2, A8; ::_thesis: verum
end;
end;
end;
hence  ((reproj (i,x)) . r) - x = (reproj (i,(0* m))) . (r - ((proj (i,m)) . x)) by A1, FINSEQ_1:13; ::_thesis: x - ((reproj (i,x)) . r) = (reproj (i,(0* m))) . (((proj (i,m)) . x) - r)
 for k being Nat st k in dom s holds
s . k = t . k
proof
let k be Nat; ::_thesis: ( k in dom s implies s . k = t . k )
assume A11: k in dom s ; ::_thesis: s . k = t . k
then A12: ( 1 <= k & k <= m ) by A1, FINSEQ_1:1;
then  k in dom (Replace (x,i,r)) by A4, FINSEQ_3:25;
then A13: (Replace (x,i,r)) /. k = (Replace (x,i,r)) . k by PARTFUN1:def_6;
A14: s . k = (x1 . k) - (y1 . k) by RVSUM_1:27;
 t = Replace ((0* m),i,(((proj (i,m)) . x) - r)) by PDIFF_1:def_5;
then A15: t . k = (Replace ((0* m),i,(((proj (i,m)) . x) - r))) /. k by A1, A11, PARTFUN1:def_6;
percases ( k = i or k <> i ) ;
supposeA16: k = i ; ::_thesis: s . k = t . k
then  ( s . k = (x1 . k) - r & t . k = ((proj (i,m)) . x) - r ) by A2, A3, A12, A13, A14, A15, FINSEQ_7:8;
hence  s . k = t . k by A16, PDIFF_1:def_1; ::_thesis: verum
end;
suppose k <> i ; ::_thesis: s . k = t . k
then  ( (Replace (x,i,r)) . k = x1 /. k & t . k = (0* m) /. k ) by A3, A12, A13, A15, FINSEQ_7:10;
then  ( (Replace (x,i,r)) . k = x1 . k & t . k = (m |-> 0) . k ) by A1, A11, PARTFUN1:def_6;
hence  s . k = t . k by A2, A14; ::_thesis: verum
end;
end;
end;
hence  x - ((reproj (i,x)) . r) = (reproj (i,(0* m))) . (((proj (i,m)) . x) - r) by A1, FINSEQ_1:13; ::_thesis: verum
end;

theorem Th7: :: PDIFF_7:7
for m, i being Element of NAT
for x being Point of (REAL-NS m)
for p being Point of (REAL-NS 1) holds
( ((reproj (i,x)) . p) - x = (reproj (i,(0. (REAL-NS m)))) . (p - ((Proj (i,m)) . x)) & x - ((reproj (i,x)) . p) = (reproj (i,(0. (REAL-NS m)))) . (((Proj (i,m)) . x) - p) )
proof
let m, i be Element of NAT ; ::_thesis: for x being Point of (REAL-NS m)
for p being Point of (REAL-NS 1) holds
( ((reproj (i,x)) . p) - x = (reproj (i,(0. (REAL-NS m)))) . (p - ((Proj (i,m)) . x)) & x - ((reproj (i,x)) . p) = (reproj (i,(0. (REAL-NS m)))) . (((Proj (i,m)) . x) - p) )
let x be Point of (REAL-NS m); ::_thesis: for p being Point of (REAL-NS 1) holds
( ((reproj (i,x)) . p) - x = (reproj (i,(0. (REAL-NS m)))) . (p - ((Proj (i,m)) . x)) & x - ((reproj (i,x)) . p) = (reproj (i,(0. (REAL-NS m)))) . (((Proj (i,m)) . x) - p) )
let p be Point of (REAL-NS 1); ::_thesis: ( ((reproj (i,x)) . p) - x = (reproj (i,(0. (REAL-NS m)))) . (p - ((Proj (i,m)) . x)) & x - ((reproj (i,x)) . p) = (reproj (i,(0. (REAL-NS m)))) . (((Proj (i,m)) . x) - p) )
consider p1 being Element of REAL , y being Element of REAL m such that
A1: ( p = <*p1*> & y = x & (reproj (i,x)) . p = (reproj (i,y)) . p1 ) by PDIFF_1:def_6;
reconsider pm = p as Element of REAL 1 by REAL_NS1:def_4;
reconsider rm = (reproj (i,y)) . p1 as Element of REAL m ;
 ((reproj (i,x)) . p) - x = rm - y by A1, REAL_NS1:5;
then A2: ((reproj (i,x)) . p) - x = (reproj (i,(0* m))) . (p1 - ((proj (i,m)) . y)) by Th6;
A3: 0* m = 0. (REAL-NS m) by REAL_NS1:def_4;
A4: <*((proj (i,m)) . y)*> = (Proj (i,m)) . x by A1, PDIFF_1:def_4;
reconsider Pr = (Proj (i,m)) . x as Element of REAL 1 by REAL_NS1:def_4;
consider r1 being Element of REAL , z being Element of REAL m such that
A5: ( p - ((Proj (i,m)) . x) = <*r1*> & z = 0. (REAL-NS m) & (reproj (i,(0. (REAL-NS m)))) . (p - ((Proj (i,m)) . x)) = (reproj (i,z)) . r1 ) by PDIFF_1:def_6;
 p - ((Proj (i,m)) . x) = pm - Pr by REAL_NS1:5;
then  p - ((Proj (i,m)) . x) = <*(p1 - ((proj (i,m)) . y))*> by A1, A4, RVSUM_1:29;
hence  ((reproj (i,x)) . p) - x = (reproj (i,(0. (REAL-NS m)))) . (p - ((Proj (i,m)) . x)) by A2, A3, A5, FINSEQ_1:76; ::_thesis: x - ((reproj (i,x)) . p) = (reproj (i,(0. (REAL-NS m)))) . (((Proj (i,m)) . x) - p)
 x - ((reproj (i,x)) . p) = y - rm by A1, REAL_NS1:5;
then A6: x - ((reproj (i,x)) . p) = (reproj (i,(0* m))) . (((proj (i,m)) . y) - p1) by Th6;
consider r2 being Element of REAL , z being Element of REAL m such that
A7: ( ((Proj (i,m)) . x) - p = <*r2*> & z = 0. (REAL-NS m) & (reproj (i,(0. (REAL-NS m)))) . (((Proj (i,m)) . x) - p) = (reproj (i,z)) . r2 ) by PDIFF_1:def_6;
 ((Proj (i,m)) . x) - p = Pr - pm by REAL_NS1:5;
then  ((Proj (i,m)) . x) - p = <*(((proj (i,m)) . y) - p1)*> by A1, A4, RVSUM_1:29;
hence  x - ((reproj (i,x)) . p) = (reproj (i,(0. (REAL-NS m)))) . (((Proj (i,m)) . x) - p) by A3, A7, A6, FINSEQ_1:76; ::_thesis: verum
end;

Lm5:  for m being Element of NAT
for nx being Point of (REAL-NS m)
for Z being Subset of (REAL-NS m)
for i being Element of NAT st Z is open & nx in Z & 1 <= i & i <= m holds
ex N being Neighbourhood of (Proj (i,m)) . nx st
for z being Point of (REAL-NS 1) st z in N holds
(reproj (i,nx)) . z in Z
proof
let m be Element of NAT ; ::_thesis: for nx being Point of (REAL-NS m)
for Z being Subset of (REAL-NS m)
for i being Element of NAT st Z is open & nx in Z & 1 <= i & i <= m holds
ex N being Neighbourhood of (Proj (i,m)) . nx st
for z being Point of (REAL-NS 1) st z in N holds
(reproj (i,nx)) . z in Z
let nx be Point of (REAL-NS m); ::_thesis: for Z being Subset of (REAL-NS m)
for i being Element of NAT st Z is open & nx in Z & 1 <= i & i <= m holds
ex N being Neighbourhood of (Proj (i,m)) . nx st
for z being Point of (REAL-NS 1) st z in N holds
(reproj (i,nx)) . z in Z
let Z be Subset of (REAL-NS m); ::_thesis: for i being Element of NAT st Z is open & nx in Z & 1 <= i & i <= m holds
ex N being Neighbourhood of (Proj (i,m)) . nx st
for z being Point of (REAL-NS 1) st z in N holds
(reproj (i,nx)) . z in Z
let i be Element of NAT ; ::_thesis: ( Z is open & nx in Z & 1 <= i & i <= m implies ex N being Neighbourhood of (Proj (i,m)) . nx st
for z being Point of (REAL-NS 1) st z in N holds
(reproj (i,nx)) . z in Z )
assume that
A1: Z is open and
A2: nx in Z and
A3: ( 1 <= i & i <= m ) ; ::_thesis: ex N being Neighbourhood of (Proj (i,m)) . nx st
for z being Point of (REAL-NS 1) st z in N holds
(reproj (i,nx)) . z in Z
consider r being Real such that
A4: ( 0 < r &  {  y where y is Point of (REAL-NS m) : ||.(y - nx).|| < r  }  c= Z ) by A1, A2, NDIFF_1:3;
set N =  {  y where y is Point of (REAL-NS 1) : ||.(y - ((Proj (i,m)) . nx)).|| < r  }  ;
reconsider N =  {  y where y is Point of (REAL-NS 1) : ||.(y - ((Proj (i,m)) . nx)).|| < r  }  as Neighbourhood of (Proj (i,m)) . nx by A4, NFCONT_1:3;
take  N ; ::_thesis: for z being Point of (REAL-NS 1) st z in N holds
(reproj (i,nx)) . z in Z
let z be Point of (REAL-NS 1); ::_thesis: ( z in N implies (reproj (i,nx)) . z in Z )
assume  z in N ; ::_thesis: (reproj (i,nx)) . z in Z
then A5: ex y being Point of (REAL-NS 1) st
( y = z & ||.(y - ((Proj (i,m)) . nx)).|| < r ) ;
||.(((reproj (i,nx)) . z) - nx).|| = ||.((reproj (i,(0. (REAL-NS m)))) . (z - ((Proj (i,m)) . nx))).|| by Th7
.= ||.(z - ((Proj (i,m)) . nx)).|| by A3, Th5 ;
then  (reproj (i,nx)) . z in  {  y where y is Point of (REAL-NS m) : ||.(y - nx).|| < r  }  by A5;
hence  (reproj (i,nx)) . z in Z by A4; ::_thesis: verum
end;

theorem Th8: :: PDIFF_7:8
for m, n being non empty Element of NAT
for i being Element of NAT
for f being PartFunc of (REAL-NS m),(REAL-NS n)
for Z being Subset of (REAL-NS m) st Z is open & 1 <= i & i <= m holds
( f is_partial_differentiable_on Z,i iff ( Z c= dom f & ( for x being Point of (REAL-NS m) st x in Z holds
f is_partial_differentiable_in x,i ) ) )
proof
let m, n be non empty Element of NAT ; ::_thesis: for i being Element of NAT
for f being PartFunc of (REAL-NS m),(REAL-NS n)
for Z being Subset of (REAL-NS m) st Z is open & 1 <= i & i <= m holds
( f is_partial_differentiable_on Z,i iff ( Z c= dom f & ( for x being Point of (REAL-NS m) st x in Z holds
f is_partial_differentiable_in x,i ) ) )
let i be Element of NAT ; ::_thesis: for f being PartFunc of (REAL-NS m),(REAL-NS n)
for Z being Subset of (REAL-NS m) st Z is open & 1 <= i & i <= m holds
( f is_partial_differentiable_on Z,i iff ( Z c= dom f & ( for x being Point of (REAL-NS m) st x in Z holds
f is_partial_differentiable_in x,i ) ) )
let f be PartFunc of (REAL-NS m),(REAL-NS n); ::_thesis: for Z being Subset of (REAL-NS m) st Z is open & 1 <= i & i <= m holds
( f is_partial_differentiable_on Z,i iff ( Z c= dom f & ( for x being Point of (REAL-NS m) st x in Z holds
f is_partial_differentiable_in x,i ) ) )
let Z be Subset of (REAL-NS m); ::_thesis: ( Z is open & 1 <= i & i <= m implies ( f is_partial_differentiable_on Z,i iff ( Z c= dom f & ( for x being Point of (REAL-NS m) st x in Z holds
f is_partial_differentiable_in x,i ) ) ) )
assume that
A1: Z is open and
A2: ( 1 <= i & i <= m ) ; ::_thesis: ( f is_partial_differentiable_on Z,i iff ( Z c= dom f & ( for x being Point of (REAL-NS m) st x in Z holds
f is_partial_differentiable_in x,i ) ) )
set S = REAL-NS 1;
set T = REAL-NS n;
set RNS = R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS n));
thus  ( f is_partial_differentiable_on Z,i implies ( Z c= dom f & ( for x being Point of (REAL-NS m) st x in Z holds
f is_partial_differentiable_in x,i ) ) ) ::_thesis: ( Z c= dom f & ( for x being Point of (REAL-NS m) st x in Z holds
f is_partial_differentiable_in x,i ) implies f is_partial_differentiable_on Z,i )
proof
assume A3: f is_partial_differentiable_on Z,i ; ::_thesis: ( Z c= dom f & ( for x being Point of (REAL-NS m) st x in Z holds
f is_partial_differentiable_in x,i ) )
hence A4: Z c= dom f by PDIFF_1:def_19; ::_thesis: for x being Point of (REAL-NS m) st x in Z holds
f is_partial_differentiable_in x,i
let nx0 be Point of (REAL-NS m); ::_thesis: ( nx0 in Z implies f is_partial_differentiable_in nx0,i )
reconsider x0 = (Proj (i,m)) . nx0 as Point of (REAL-NS 1) ;
assume A5: nx0 in Z ; ::_thesis: f is_partial_differentiable_in nx0,i
then  f | Z is_partial_differentiable_in nx0,i by A3, PDIFF_1:def_19;
then  (f | Z) * (reproj (i,nx0)) is_differentiable_in x0 by PDIFF_1:def_9;
then consider N0 being Neighbourhood of x0 such that
A6: N0 c= dom ((f | Z) * (reproj (i,nx0))) and
A7: ex L being Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))) ex R being RestFunc of (REAL-NS 1),(REAL-NS n) st
for x being Point of (REAL-NS 1) st x in N0 holds
(((f | Z) * (reproj (i,nx0))) /. x) - (((f | Z) * (reproj (i,nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0)) by NDIFF_1:def_6;
consider L being Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))), R being RestFunc of (REAL-NS 1),(REAL-NS n) such that
A8: for x being Point of (REAL-NS 1) st x in N0 holds
(((f | Z) * (reproj (i,nx0))) /. x) - (((f | Z) * (reproj (i,nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0)) by A7;
consider N1 being Neighbourhood of x0 such that
A9: for x being Point of (REAL-NS 1) st x in N1 holds
(reproj (i,nx0)) . x in Z by A1, A2, A5, Lm5;
A10: now__::_thesis:_for_x_being_Point_of_(REAL-NS_1)_st_x_in_N1_holds_
(reproj_(i,nx0))_._x_in_dom_(f_|_Z)
let x be Point of (REAL-NS 1); ::_thesis: ( x in N1 implies (reproj (i,nx0)) . x in dom (f | Z) )
assume  x in N1 ; ::_thesis: (reproj (i,nx0)) . x in dom (f | Z)
then  (reproj (i,nx0)) . x in Z by A9;
then  (reproj (i,nx0)) . x in (dom f) /\ Z by A4, XBOOLE_0:def_4;
hence  (reproj (i,nx0)) . x in dom (f | Z) by RELAT_1:61; ::_thesis: verum
end;
reconsider N = N0 /\ N1 as Neighbourhood of x0 by Lm1;
 (f | Z) * (reproj (i,nx0)) c= f * (reproj (i,nx0)) by RELAT_1:29, RELAT_1:59;
then A11: dom ((f | Z) * (reproj (i,nx0))) c= dom (f * (reproj (i,nx0))) by RELAT_1:11;
 N c= N0 by XBOOLE_1:17;
then  N c= dom ((f | Z) * (reproj (i,nx0))) by A6, XBOOLE_1:1;
then A12: N c= dom (f * (reproj (i,nx0))) by A11, XBOOLE_1:1;
now__::_thesis:_for_x_being_Point_of_(REAL-NS_1)_st_x_in_N_holds_
((f_*_(reproj_(i,nx0)))_/._x)_-_((f_*_(reproj_(i,nx0)))_/._x0)_=_(L_._(x_-_x0))_+_(R_/._(x_-_x0))
let x be Point of (REAL-NS 1); ::_thesis: ( x in N implies ((f * (reproj (i,nx0))) /. x) - ((f * (reproj (i,nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0)) )
assume A13: x in N ; ::_thesis: ((f * (reproj (i,nx0))) /. x) - ((f * (reproj (i,nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0))
then A14: x in N0 by XBOOLE_0:def_4;
A15: dom (reproj (i,nx0)) = the carrier of (REAL-NS 1) by FUNCT_2:def_1;
 x in N1 by A13, XBOOLE_0:def_4;
then A16: (reproj (i,nx0)) . x in dom (f | Z) by A10;
then A17: ( (reproj (i,nx0)) . x in dom f & (reproj (i,nx0)) . x in Z ) by RELAT_1:57;
A18: (reproj (i,nx0)) . x0 in dom (f | Z) by A10, NFCONT_1:4;
then A19: ( (reproj (i,nx0)) . x0 in dom f & (reproj (i,nx0)) . x0 in Z ) by RELAT_1:57;
A20: ((f | Z) * (reproj (i,nx0))) /. x = (f | Z) /. ((reproj (i,nx0)) /. x) by A16, A15, PARTFUN2:4
.= f /. ((reproj (i,nx0)) /. x) by A17, PARTFUN2:17
.= (f * (reproj (i,nx0))) /. x by A15, A17, PARTFUN2:4 ;
((f | Z) * (reproj (i,nx0))) /. x0 = (f | Z) /. ((reproj (i,nx0)) /. x0) by A15, A18, PARTFUN2:4
.= f /. ((reproj (i,nx0)) /. x0) by A19, PARTFUN2:17
.= (f * (reproj (i,nx0))) /. x0 by A15, A19, PARTFUN2:4 ;
hence  ((f * (reproj (i,nx0))) /. x) - ((f * (reproj (i,nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0)) by A8, A14, A20; ::_thesis: verum
end;
then  f * (reproj (i,nx0)) is_differentiable_in x0 by A12, NDIFF_1:def_6;
hence  f is_partial_differentiable_in nx0,i by PDIFF_1:def_9; ::_thesis: verum
end;
assume that
A21: Z c= dom f and
A22: for nx being Point of (REAL-NS m) st nx in Z holds
f is_partial_differentiable_in nx,i ; ::_thesis: f is_partial_differentiable_on Z,i
thus  Z c= dom f by A21; :: according to PDIFF_1:def_19 ::_thesis: for b1 being Element of the carrier of (REAL-NS m) holds
( not b1 in Z or f | Z is_partial_differentiable_in b1,i )
now__::_thesis:_for_nx0_being_Point_of_(REAL-NS_m)_st_nx0_in_Z_holds_
f_|_Z_is_partial_differentiable_in_nx0,i
let nx0 be Point of (REAL-NS m); ::_thesis: ( nx0 in Z implies f | Z is_partial_differentiable_in nx0,i )
assume A23: nx0 in Z ; ::_thesis: f | Z is_partial_differentiable_in nx0,i
then A24: f is_partial_differentiable_in nx0,i by A22;
reconsider x0 = (Proj (i,m)) . nx0 as Point of (REAL-NS 1) ;
 f * (reproj (i,nx0)) is_differentiable_in x0 by A24, PDIFF_1:def_9;
then consider N0 being Neighbourhood of x0 such that
 N0 c= dom (f * (reproj (i,nx0))) and
A25: ex L being Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))) ex R being RestFunc of (REAL-NS 1),(REAL-NS n) st
for x being Point of (REAL-NS 1) st x in N0 holds
((f * (reproj (i,nx0))) /. x) - ((f * (reproj (i,nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0)) by NDIFF_1:def_6;
consider N1 being Neighbourhood of x0 such that
A26: for x being Point of (REAL-NS 1) st x in N1 holds
(reproj (i,nx0)) . x in Z by A1, A2, A23, Lm5;
A27: now__::_thesis:_for_x_being_Point_of_(REAL-NS_1)_st_x_in_N1_holds_
(reproj_(i,nx0))_._x_in_dom_(f_|_Z)
let x be Point of (REAL-NS 1); ::_thesis: ( x in N1 implies (reproj (i,nx0)) . x in dom (f | Z) )
assume  x in N1 ; ::_thesis: (reproj (i,nx0)) . x in dom (f | Z)
then  (reproj (i,nx0)) . x in Z by A26;
then  (reproj (i,nx0)) . x in (dom f) /\ Z by A21, XBOOLE_0:def_4;
hence  (reproj (i,nx0)) . x in dom (f | Z) by RELAT_1:61; ::_thesis: verum
end;
A28: N1 c= dom ((f | Z) * (reproj (i,nx0)))
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in N1 or z in dom ((f | Z) * (reproj (i,nx0))) )
assume A29: z in N1 ; ::_thesis: z in dom ((f | Z) * (reproj (i,nx0)))
then A30: z in the carrier of (REAL-NS 1) ;
reconsider x = z as Point of (REAL-NS 1) by A29;
A31: (reproj (i,nx0)) . x in dom (f | Z) by A29, A27;
 z in dom (reproj (i,nx0)) by A30, FUNCT_2:def_1;
hence  z in dom ((f | Z) * (reproj (i,nx0))) by A31, FUNCT_1:11; ::_thesis: verum
end;
reconsider N = N0 /\ N1 as Neighbourhood of x0 by Lm1;
 N c= N1 by XBOOLE_1:17;
then A32: N c= dom ((f | Z) * (reproj (i,nx0))) by A28, XBOOLE_1:1;
consider L being Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))), R being RestFunc of (REAL-NS 1),(REAL-NS n) such that
A33: for x being Point of (REAL-NS 1) st x in N0 holds
((f * (reproj (i,nx0))) /. x) - ((f * (reproj (i,nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0)) by A25;
now__::_thesis:_for_x_being_Point_of_(REAL-NS_1)_st_x_in_N_holds_
(((f_|_Z)_*_(reproj_(i,nx0)))_/._x)_-_(((f_|_Z)_*_(reproj_(i,nx0)))_/._x0)_=_(L_._(x_-_x0))_+_(R_/._(x_-_x0))
let x be Point of (REAL-NS 1); ::_thesis: ( x in N implies (((f | Z) * (reproj (i,nx0))) /. x) - (((f | Z) * (reproj (i,nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0)) )
assume A34: x in N ; ::_thesis: (((f | Z) * (reproj (i,nx0))) /. x) - (((f | Z) * (reproj (i,nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0))
then A35: x in N0 by XBOOLE_0:def_4;
A36: dom (reproj (i,nx0)) = the carrier of (REAL-NS 1) by FUNCT_2:def_1;
 x in N1 by A34, XBOOLE_0:def_4;
then A37: (reproj (i,nx0)) . x in dom (f | Z) by A27;
then A38: (reproj (i,nx0)) . x in (dom f) /\ Z by RELAT_1:61;
then A39: (reproj (i,nx0)) . x in dom f by XBOOLE_0:def_4;
A40: (reproj (i,nx0)) . x0 in dom (f | Z) by A27, NFCONT_1:4;
then A41: (reproj (i,nx0)) . x0 in (dom f) /\ Z by RELAT_1:61;
then A42: (reproj (i,nx0)) . x0 in dom f by XBOOLE_0:def_4;
A43: ((f | Z) * (reproj (i,nx0))) /. x = (f | Z) /. ((reproj (i,nx0)) /. x) by A37, A36, PARTFUN2:4
.= f /. ((reproj (i,nx0)) /. x) by A38, PARTFUN2:16
.= (f * (reproj (i,nx0))) /. x by A36, A39, PARTFUN2:4 ;
((f | Z) * (reproj (i,nx0))) /. x0 = (f | Z) /. ((reproj (i,nx0)) /. x0) by A36, A40, PARTFUN2:4
.= f /. ((reproj (i,nx0)) /. x0) by A41, PARTFUN2:16
.= (f * (reproj (i,nx0))) /. x0 by A36, A42, PARTFUN2:4 ;
hence  (((f | Z) * (reproj (i,nx0))) /. x) - (((f | Z) * (reproj (i,nx0))) /. x0) = (L . (x - x0)) + (R /. (x - x0)) by A43, A35, A33; ::_thesis: verum
end;
then  (f | Z) * (reproj (i,nx0)) is_differentiable_in x0 by A32, NDIFF_1:def_6;
hence  f | Z is_partial_differentiable_in nx0,i by PDIFF_1:def_9; ::_thesis: verum
end;
hence  for b1 being Element of the carrier of (REAL-NS m) holds
( not b1 in Z or f | Z is_partial_differentiable_in b1,i ) ; ::_thesis: verum
end;

Lm6:  for m being non empty Element of NAT
for v being Element of REAL m
for x being Element of REAL
for i being Element of NAT holds len (Replace (v,i,x)) = m
proof
let m be non empty Element of NAT ; ::_thesis: for v being Element of REAL m
for x being Element of REAL
for i being Element of NAT holds len (Replace (v,i,x)) = m
let v be Element of REAL m; ::_thesis: for x being Element of REAL
for i being Element of NAT holds len (Replace (v,i,x)) = m
let x be Element of REAL ; ::_thesis: for i being Element of NAT holds len (Replace (v,i,x)) = m
let i be Element of NAT ; ::_thesis: len (Replace (v,i,x)) = m
 len (Replace (v,i,x)) = len v by FUNCT_7:97;
hence  len (Replace (v,i,x)) = m by CARD_1:def_7; ::_thesis: verum
end;

Lm7:  for m being non empty Element of NAT
for x being Element of REAL
for i, j being Element of NAT st 1 <= j & j <= m holds
( ( i = j implies (Replace ((0* m),i,x)) . j = x ) & ( i <> j implies (Replace ((0* m),i,x)) . j = 0 ) )
proof
let m be non empty Element of NAT ; ::_thesis: for x being Element of REAL
for i, j being Element of NAT st 1 <= j & j <= m holds
( ( i = j implies (Replace ((0* m),i,x)) . j = x ) & ( i <> j implies (Replace ((0* m),i,x)) . j = 0 ) )
let x be Element of REAL ; ::_thesis: for i, j being Element of NAT st 1 <= j & j <= m holds
( ( i = j implies (Replace ((0* m),i,x)) . j = x ) & ( i <> j implies (Replace ((0* m),i,x)) . j = 0 ) )
let i, j be Element of NAT ; ::_thesis: ( 1 <= j & j <= m implies ( ( i = j implies (Replace ((0* m),i,x)) . j = x ) & ( i <> j implies (Replace ((0* m),i,x)) . j = 0 ) ) )
assume  ( 1 <= j & j <= m ) ; ::_thesis: ( ( i = j implies (Replace ((0* m),i,x)) . j = x ) & ( i <> j implies (Replace ((0* m),i,x)) . j = 0 ) )
then A1: j in Seg m ;
 len (0* m) = m by CARD_1:def_7;
then A2: j in dom (0* m) by A1, FINSEQ_1:def_3;
now__::_thesis:_(_i_<>_j_implies_(Replace_((0*_m),i,x))_._j_=_0_)
assume  i <> j ; ::_thesis: (Replace ((0* m),i,x)) . j = 0
then  ((0* m) +* (i,x)) . j = (0* m) . j by FUNCT_7:32;
hence  (Replace ((0* m),i,x)) . j = 0 ; ::_thesis: verum
end;
hence  ( ( i = j implies (Replace ((0* m),i,x)) . j = x ) & ( i <> j implies (Replace ((0* m),i,x)) . j = 0 ) ) by A2, FUNCT_7:31; ::_thesis: verum
end;

theorem Th9: :: PDIFF_7:9
for m being non empty Element of NAT
for x, y being Element of REAL
for i being Element of NAT st 1 <= i & i <= m holds
Replace ((0* m),i,(x + y)) = (Replace ((0* m),i,x)) + (Replace ((0* m),i,y))
proof
let m be non empty Element of NAT ; ::_thesis: for x, y being Element of REAL
for i being Element of NAT st 1 <= i & i <= m holds
Replace ((0* m),i,(x + y)) = (Replace ((0* m),i,x)) + (Replace ((0* m),i,y))
let x, y be Element of REAL ; ::_thesis: for i being Element of NAT st 1 <= i & i <= m holds
Replace ((0* m),i,(x + y)) = (Replace ((0* m),i,x)) + (Replace ((0* m),i,y))
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m implies Replace ((0* m),i,(x + y)) = (Replace ((0* m),i,x)) + (Replace ((0* m),i,y)) )
assume A1: ( 1 <= i & i <= m ) ; ::_thesis: Replace ((0* m),i,(x + y)) = (Replace ((0* m),i,x)) + (Replace ((0* m),i,y))
A2: ( len (Replace ((0* m),i,(x + y))) = m & len (Replace ((0* m),i,x)) = m & len (Replace ((0* m),i,y)) = m ) by Lm6;
then A3: len ((Replace ((0* m),i,x)) + (Replace ((0* m),i,y))) = len (Replace ((0* m),i,(x + y))) by RVSUM_1:115;
 for j being Nat st 1 <= j & j <= len (Replace ((0* m),i,(x + y))) holds
(Replace ((0* m),i,(x + y))) . j = ((Replace ((0* m),i,x)) + (Replace ((0* m),i,y))) . j
proof
let j be Nat; ::_thesis: ( 1 <= j & j <= len (Replace ((0* m),i,(x + y))) implies (Replace ((0* m),i,(x + y))) . j = ((Replace ((0* m),i,x)) + (Replace ((0* m),i,y))) . j )
assume A4: ( 1 <= j & j <= len (Replace ((0* m),i,(x + y))) ) ; ::_thesis: (Replace ((0* m),i,(x + y))) . j = ((Replace ((0* m),i,x)) + (Replace ((0* m),i,y))) . j
reconsider j = j as Element of NAT by ORDINAL1:def_12;
A5: dom ((Replace ((0* m),i,x)) + (Replace ((0* m),i,y))) = (dom (Replace ((0* m),i,x))) /\ (dom (Replace ((0* m),i,y))) by VALUED_1:def_1;
 ( j in dom (Replace ((0* m),i,x)) & j in dom (Replace ((0* m),i,y)) ) by A4, A2, FINSEQ_3:25;
then  j in dom ((Replace ((0* m),i,x)) + (Replace ((0* m),i,y))) by A5, XBOOLE_0:def_4;
then A6: ((Replace ((0* m),i,x)) + (Replace ((0* m),i,y))) . j = ((Replace ((0* m),i,x)) . j) + ((Replace ((0* m),i,y)) . j) by VALUED_1:def_1;
percases ( i = j or i <> j ) ;
supposeA7: i = j ; ::_thesis: (Replace ((0* m),i,(x + y))) . j = ((Replace ((0* m),i,x)) + (Replace ((0* m),i,y))) . j
then ((Replace ((0* m),i,x)) + (Replace ((0* m),i,y))) . j = x + ((Replace ((0* m),i,y)) . j) by A1, A6, Lm7
.= x + y by A1, A7, Lm7 ;
hence  (Replace ((0* m),i,(x + y))) . j = ((Replace ((0* m),i,x)) + (Replace ((0* m),i,y))) . j by A1, A7, Lm7; ::_thesis: verum
end;
supposeA8: i <> j ; ::_thesis: (Replace ((0* m),i,(x + y))) . j = ((Replace ((0* m),i,x)) + (Replace ((0* m),i,y))) . j
then ((Replace ((0* m),i,x)) + (Replace ((0* m),i,y))) . j = 0 + ((Replace ((0* m),i,y)) . j) by A4, A6, A2, Lm7
.= 0 by A4, A2, A8, Lm7 ;
hence  (Replace ((0* m),i,(x + y))) . j = ((Replace ((0* m),i,x)) + (Replace ((0* m),i,y))) . j by A4, A2, A8, Lm7; ::_thesis: verum
end;
end;
end;
hence  Replace ((0* m),i,(x + y)) = (Replace ((0* m),i,x)) + (Replace ((0* m),i,y)) by A3, FINSEQ_1:14; ::_thesis: verum
end;

theorem Th10: :: PDIFF_7:10
for m being non empty Element of NAT
for x, a being Element of REAL
for i being Element of NAT st 1 <= i & i <= m holds
Replace ((0* m),i,(a * x)) = a * (Replace ((0* m),i,x))
proof
let m be non empty Element of NAT ; ::_thesis: for x, a being Element of REAL
for i being Element of NAT st 1 <= i & i <= m holds
Replace ((0* m),i,(a * x)) = a * (Replace ((0* m),i,x))
let x, a be Element of REAL ; ::_thesis: for i being Element of NAT st 1 <= i & i <= m holds
Replace ((0* m),i,(a * x)) = a * (Replace ((0* m),i,x))
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m implies Replace ((0* m),i,(a * x)) = a * (Replace ((0* m),i,x)) )
assume A1: ( 1 <= i & i <= m ) ; ::_thesis: Replace ((0* m),i,(a * x)) = a * (Replace ((0* m),i,x))
A2: ( len (Replace ((0* m),i,(a * x))) = m & len (Replace ((0* m),i,x)) = m ) by Lm6;
then A3: len (a * (Replace ((0* m),i,x))) = len (Replace ((0* m),i,(a * x))) by RVSUM_1:117;
 for j being Nat st 1 <= j & j <= len (Replace ((0* m),i,(a * x))) holds
(Replace ((0* m),i,(a * x))) . j = (a * (Replace ((0* m),i,x))) . j
proof
let j be Nat; ::_thesis: ( 1 <= j & j <= len (Replace ((0* m),i,(a * x))) implies (Replace ((0* m),i,(a * x))) . j = (a * (Replace ((0* m),i,x))) . j )
assume A4: ( 1 <= j & j <= len (Replace ((0* m),i,(a * x))) ) ; ::_thesis: (Replace ((0* m),i,(a * x))) . j = (a * (Replace ((0* m),i,x))) . j
reconsider j = j as Element of NAT by ORDINAL1:def_12;
percases ( i = j or i <> j ) ;
supposeA5: i = j ; ::_thesis: (Replace ((0* m),i,(a * x))) . j = (a * (Replace ((0* m),i,x))) . j
then (Replace ((0* m),i,(a * x))) . j = a * x by A1, Lm7
.= a * ((Replace ((0* m),i,x)) . j) by A1, A5, Lm7 ;
hence  (Replace ((0* m),i,(a * x))) . j = (a * (Replace ((0* m),i,x))) . j by RVSUM_1:44; ::_thesis: verum
end;
supposeA6: i <> j ; ::_thesis: (Replace ((0* m),i,(a * x))) . j = (a * (Replace ((0* m),i,x))) . j
then  (Replace ((0* m),i,x)) . j = 0 by A2, A4, Lm7;
then  (Replace ((0* m),i,(a * x))) . j = a * ((Replace ((0* m),i,x)) . j) by A2, A4, A6, Lm7;
hence  (Replace ((0* m),i,(a * x))) . j = (a * (Replace ((0* m),i,x))) . j by RVSUM_1:44; ::_thesis: verum
end;
end;
end;
hence  Replace ((0* m),i,(a * x)) = a * (Replace ((0* m),i,x)) by A3, FINSEQ_1:14; ::_thesis: verum
end;

theorem Th11: :: PDIFF_7:11
for m being non empty Element of NAT
for x being Element of REAL
for i being Element of NAT st 1 <= i & i <= m & x <> 0 holds
Replace ((0* m),i,x) <> 0* m
proof
let m be non empty Element of NAT ; ::_thesis: for x being Element of REAL
for i being Element of NAT st 1 <= i & i <= m & x <> 0 holds
Replace ((0* m),i,x) <> 0* m
let x be Element of REAL ; ::_thesis: for i being Element of NAT st 1 <= i & i <= m & x <> 0 holds
Replace ((0* m),i,x) <> 0* m
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m & x <> 0 implies Replace ((0* m),i,x) <> 0* m )
assume A1: ( 1 <= i & i <= m & x <> 0 ) ; ::_thesis: Replace ((0* m),i,x) <> 0* m
then A2: i in Seg m ;
assume A3: Replace ((0* m),i,x) = 0* m ; ::_thesis: contradiction
 len (0* m) = m by CARD_1:def_7;
then  Seg m = proj1 (0* m) by FINSEQ_1:def_3;
then  x = (0* m) . i by A3, A2, FUNCT_7:31;
hence  contradiction by A1; ::_thesis: verum
end;

theorem Th12: :: PDIFF_7:12
for m being non empty Element of NAT
for x, y being Element of REAL
for z being Element of REAL m
for i being Element of NAT st 1 <= i & i <= m & y = (proj (i,m)) . z holds
( (Replace (z,i,x)) - z = Replace ((0* m),i,(x - y)) & z - (Replace (z,i,x)) = Replace ((0* m),i,(y - x)) )
proof
let m be non empty Element of NAT ; ::_thesis: for x, y being Element of REAL
for z being Element of REAL m
for i being Element of NAT st 1 <= i & i <= m & y = (proj (i,m)) . z holds
( (Replace (z,i,x)) - z = Replace ((0* m),i,(x - y)) & z - (Replace (z,i,x)) = Replace ((0* m),i,(y - x)) )
let x, y be Element of REAL ; ::_thesis: for z being Element of REAL m
for i being Element of NAT st 1 <= i & i <= m & y = (proj (i,m)) . z holds
( (Replace (z,i,x)) - z = Replace ((0* m),i,(x - y)) & z - (Replace (z,i,x)) = Replace ((0* m),i,(y - x)) )
let z be Element of REAL m; ::_thesis: for i being Element of NAT st 1 <= i & i <= m & y = (proj (i,m)) . z holds
( (Replace (z,i,x)) - z = Replace ((0* m),i,(x - y)) & z - (Replace (z,i,x)) = Replace ((0* m),i,(y - x)) )
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m & y = (proj (i,m)) . z implies ( (Replace (z,i,x)) - z = Replace ((0* m),i,(x - y)) & z - (Replace (z,i,x)) = Replace ((0* m),i,(y - x)) ) )
assume that
A1: ( 1 <= i & i <= m ) and
A2: y = (proj (i,m)) . z ; ::_thesis: ( (Replace (z,i,x)) - z = Replace ((0* m),i,(x - y)) & z - (Replace (z,i,x)) = Replace ((0* m),i,(y - x)) )
A3: ( len (Replace ((0* m),i,(x - y))) = m & len (Replace ((0* m),i,x)) = m & len (Replace ((0* m),i,(- y))) = m ) by Lm6;
A4: dom (Replace (z,i,x)) = dom z by FUNCT_7:30;
A5: ( dom (- z) = dom z & dom (- (Replace (z,i,x))) = dom (Replace (z,i,x)) ) by VALUED_1:def_5;
A6: dom ((Replace (z,i,x)) - z) = (dom (Replace (z,i,x))) /\ (dom (- z)) by VALUED_1:def_1;
A7: len (0* m) = m by CARD_1:def_7;
 dom ((Replace (z,i,x)) - z) = Seg m by A4, A5, A6, FINSEQ_1:89;
then  len ((Replace (z,i,x)) - z) = len (0* m) by A7, FINSEQ_1:def_3;
then A8: len ((Replace (z,i,x)) - z) = len (Replace ((0* m),i,(x - y))) by FINSEQ_7:5;
 for j being Nat st 1 <= j & j <= len ((Replace (z,i,x)) - z) holds
(Replace ((0* m),i,(x - y))) . j = ((Replace (z,i,x)) - z) . j
proof
let j be Nat; ::_thesis: ( 1 <= j & j <= len ((Replace (z,i,x)) - z) implies (Replace ((0* m),i,(x - y))) . j = ((Replace (z,i,x)) - z) . j )
assume A9: ( 1 <= j & j <= len ((Replace (z,i,x)) - z) ) ; ::_thesis: (Replace ((0* m),i,(x - y))) . j = ((Replace (z,i,x)) - z) . j
reconsider j = j as Element of NAT by ORDINAL1:def_12;
A10: j in dom ((Replace (z,i,x)) - z) by A9, FINSEQ_3:25;
 (- z) . j = (- 1) * (z . j) by RVSUM_1:44;
then  ((Replace (z,i,x)) - z) . j = ((Replace (z,i,x)) . j) + (- (z . j)) by A10, VALUED_1:def_1;
then A11: ((Replace (z,i,x)) - z) . j = ((Replace (z,i,x)) . j) - (z . j) ;
A12: ( 1 <= i & i <= len z implies (Replace (z,i,x)) . i = x )
proof
assume  ( 1 <= i & i <= len z ) ; ::_thesis: (Replace (z,i,x)) . i = x
then  i in dom z by FINSEQ_3:25;
hence  (Replace (z,i,x)) . i = x by FUNCT_7:31; ::_thesis: verum
end;
A13: dom ((Replace ((0* m),i,x)) + (Replace ((0* m),i,(- y)))) = (dom (Replace ((0* m),i,x))) /\ (dom (Replace ((0* m),i,(- y)))) by VALUED_1:def_1;
 ( j in dom (Replace ((0* m),i,x)) & j in dom (Replace ((0* m),i,(- y))) ) by A3, A9, A8, FINSEQ_3:25;
then  j in dom ((Replace ((0* m),i,x)) + (Replace ((0* m),i,(- y)))) by A13, XBOOLE_0:def_4;
then A14: ((Replace ((0* m),i,x)) + (Replace ((0* m),i,(- y)))) . j = ((Replace ((0* m),i,x)) . j) + ((Replace ((0* m),i,(- y))) . j) by VALUED_1:def_1;
percases ( i = j or not i = j ) ;
supposeA15: i = j ; ::_thesis: (Replace ((0* m),i,(x - y))) . j = ((Replace (z,i,x)) - z) . j
(Replace ((0* m),i,(x - y))) . j = (Replace ((0* m),i,(x + (- y)))) . j
.= ((Replace ((0* m),i,x)) + (Replace ((0* m),i,(- y)))) . j by A1, Th9
.= x + ((Replace ((0* m),i,(- y))) . j) by A1, A14, A15, Lm7
.= x + (- y) by A1, A15, Lm7
.= x - ((proj (i,m)) . z) by A2 ;
hence  (Replace ((0* m),i,(x - y))) . j = ((Replace (z,i,x)) - z) . j by A11, A9, A4, A5, A6, A12, A15, FINSEQ_3:29, PDIFF_1:def_1; ::_thesis: verum
end;
supposeA16: not i = j ; ::_thesis: (Replace ((0* m),i,(x - y))) . j = ((Replace (z,i,x)) - z) . j
then  (Replace ((0* m),i,(x - y))) . j = (z . j) - (z . j) by A3, A9, A8, Lm7;
hence  (Replace ((0* m),i,(x - y))) . j = ((Replace (z,i,x)) - z) . j by A11, A16, FUNCT_7:32; ::_thesis: verum
end;
end;
end;
hence A17: (Replace (z,i,x)) - z = Replace ((0* m),i,(x - y)) by A8, FINSEQ_1:14; ::_thesis: z - (Replace (z,i,x)) = Replace ((0* m),i,(y - x))
reconsider a = - 1 as Element of REAL by XREAL_0:def_1;
 z - (Replace (z,i,x)) = - (Replace ((0* m),i,(x - y))) by A17, RVSUM_1:35;
then  z - (Replace (z,i,x)) = Replace ((0* m),i,(a * (x - y))) by A1, Th10;
hence  z - (Replace (z,i,x)) = Replace ((0* m),i,(y - x)) ; ::_thesis: verum
end;

theorem Th13: :: PDIFF_7:13
for m being non empty Element of NAT
for x, y being Element of REAL
for i being Element of NAT st 1 <= i & i <= m holds
(reproj (i,(0* m))) . (x + y) = ((reproj (i,(0* m))) . x) + ((reproj (i,(0* m))) . y)
proof
let m be non empty Element of NAT ; ::_thesis: for x, y being Element of REAL
for i being Element of NAT st 1 <= i & i <= m holds
(reproj (i,(0* m))) . (x + y) = ((reproj (i,(0* m))) . x) + ((reproj (i,(0* m))) . y)
let x, y be Element of REAL ; ::_thesis: for i being Element of NAT st 1 <= i & i <= m holds
(reproj (i,(0* m))) . (x + y) = ((reproj (i,(0* m))) . x) + ((reproj (i,(0* m))) . y)
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m implies (reproj (i,(0* m))) . (x + y) = ((reproj (i,(0* m))) . x) + ((reproj (i,(0* m))) . y) )
assume A1: ( 1 <= i & i <= m ) ; ::_thesis: (reproj (i,(0* m))) . (x + y) = ((reproj (i,(0* m))) . x) + ((reproj (i,(0* m))) . y)
 ( Replace ((0* m),i,x) = (reproj (i,(0* m))) . x & Replace ((0* m),i,y) = (reproj (i,(0* m))) . y & (reproj (i,(0* m))) . (x + y) = Replace ((0* m),i,(x + y)) ) by PDIFF_1:def_5;
hence  (reproj (i,(0* m))) . (x + y) = ((reproj (i,(0* m))) . x) + ((reproj (i,(0* m))) . y) by A1, Th9; ::_thesis: verum
end;

theorem Th14: :: PDIFF_7:14
for m being non empty Element of NAT
for x, y being Point of (REAL-NS 1)
for i being Element of NAT st 1 <= i & i <= m holds
(reproj (i,(0. (REAL-NS m)))) . (x + y) = ((reproj (i,(0. (REAL-NS m)))) . x) + ((reproj (i,(0. (REAL-NS m)))) . y)
proof
let m be non empty Element of NAT ; ::_thesis: for x, y being Point of (REAL-NS 1)
for i being Element of NAT st 1 <= i & i <= m holds
(reproj (i,(0. (REAL-NS m)))) . (x + y) = ((reproj (i,(0. (REAL-NS m)))) . x) + ((reproj (i,(0. (REAL-NS m)))) . y)
let x, y be Point of (REAL-NS 1); ::_thesis: for i being Element of NAT st 1 <= i & i <= m holds
(reproj (i,(0. (REAL-NS m)))) . (x + y) = ((reproj (i,(0. (REAL-NS m)))) . x) + ((reproj (i,(0. (REAL-NS m)))) . y)
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m implies (reproj (i,(0. (REAL-NS m)))) . (x + y) = ((reproj (i,(0. (REAL-NS m)))) . x) + ((reproj (i,(0. (REAL-NS m)))) . y) )
assume A1: ( 1 <= i & i <= m ) ; ::_thesis: (reproj (i,(0. (REAL-NS m)))) . (x + y) = ((reproj (i,(0. (REAL-NS m)))) . x) + ((reproj (i,(0. (REAL-NS m)))) . y)
consider q1 being Element of REAL , z1 being Element of REAL m such that
A2: ( x = <*q1*> & z1 = 0. (REAL-NS m) & (reproj (i,(0. (REAL-NS m)))) . x = (reproj (i,z1)) . q1 ) by PDIFF_1:def_6;
consider q2 being Element of REAL , z2 being Element of REAL m such that
A3: ( y = <*q2*> & z2 = 0. (REAL-NS m) & (reproj (i,(0. (REAL-NS m)))) . y = (reproj (i,z2)) . q2 ) by PDIFF_1:def_6;
consider q12 being Element of REAL , z12 being Element of REAL m such that
A4: ( x + y = <*q12*> & z12 = 0. (REAL-NS m) & (reproj (i,(0. (REAL-NS m)))) . (x + y) = (reproj (i,z12)) . q12 ) by PDIFF_1:def_6;
A5: 0. (REAL-NS m) = 0* m by REAL_NS1:def_4;
reconsider qq1 = <*q1*> as Element of REAL 1 by FINSEQ_2:98;
reconsider qq2 = <*q2*> as Element of REAL 1 by FINSEQ_2:98;
 x + y = qq1 + qq2 by A2, A3, REAL_NS1:2;
then A6: x + y = <*(q1 + q2)*> by RVSUM_1:13;
((reproj (i,(0. (REAL-NS m)))) . x) + ((reproj (i,(0. (REAL-NS m)))) . y) = ((reproj (i,(0* m))) . q1) + ((reproj (i,(0* m))) . q2) by A2, A3, A5, REAL_NS1:2
.= (reproj (i,(0* m))) . (q1 + q2) by A1, Th13 ;
hence  (reproj (i,(0. (REAL-NS m)))) . (x + y) = ((reproj (i,(0. (REAL-NS m)))) . x) + ((reproj (i,(0. (REAL-NS m)))) . y) by A6, A4, A5, FINSEQ_1:76; ::_thesis: verum
end;

theorem Th15: :: PDIFF_7:15
for m being non empty Element of NAT
for x, a being Element of REAL
for i being Element of NAT st 1 <= i & i <= m holds
(reproj (i,(0* m))) . (a * x) = a * ((reproj (i,(0* m))) . x)
proof
let m be non empty Element of NAT ; ::_thesis: for x, a being Element of REAL
for i being Element of NAT st 1 <= i & i <= m holds
(reproj (i,(0* m))) . (a * x) = a * ((reproj (i,(0* m))) . x)
let x, a be Element of REAL ; ::_thesis: for i being Element of NAT st 1 <= i & i <= m holds
(reproj (i,(0* m))) . (a * x) = a * ((reproj (i,(0* m))) . x)
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m implies (reproj (i,(0* m))) . (a * x) = a * ((reproj (i,(0* m))) . x) )
assume  ( 1 <= i & i <= m ) ; ::_thesis: (reproj (i,(0* m))) . (a * x) = a * ((reproj (i,(0* m))) . x)
then A1: Replace ((0* m),i,(a * x)) = a * (Replace ((0* m),i,x)) by Th10;
 Replace ((0* m),i,x) = (reproj (i,(0* m))) . x by PDIFF_1:def_5;
hence  (reproj (i,(0* m))) . (a * x) = a * ((reproj (i,(0* m))) . x) by A1, PDIFF_1:def_5; ::_thesis: verum
end;

theorem Th16: :: PDIFF_7:16
for m being non empty Element of NAT
for x being Point of (REAL-NS 1)
for a being Element of REAL
for i being Element of NAT st 1 <= i & i <= m holds
(reproj (i,(0. (REAL-NS m)))) . (a * x) = a * ((reproj (i,(0. (REAL-NS m)))) . x)
proof
let m be non empty Element of NAT ; ::_thesis: for x being Point of (REAL-NS 1)
for a being Element of REAL
for i being Element of NAT st 1 <= i & i <= m holds
(reproj (i,(0. (REAL-NS m)))) . (a * x) = a * ((reproj (i,(0. (REAL-NS m)))) . x)
let x be Point of (REAL-NS 1); ::_thesis: for a being Element of REAL
for i being Element of NAT st 1 <= i & i <= m holds
(reproj (i,(0. (REAL-NS m)))) . (a * x) = a * ((reproj (i,(0. (REAL-NS m)))) . x)
let a be Element of REAL ; ::_thesis: for i being Element of NAT st 1 <= i & i <= m holds
(reproj (i,(0. (REAL-NS m)))) . (a * x) = a * ((reproj (i,(0. (REAL-NS m)))) . x)
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m implies (reproj (i,(0. (REAL-NS m)))) . (a * x) = a * ((reproj (i,(0. (REAL-NS m)))) . x) )
assume A1: ( 1 <= i & i <= m ) ; ::_thesis: (reproj (i,(0. (REAL-NS m)))) . (a * x) = a * ((reproj (i,(0. (REAL-NS m)))) . x)
consider q1 being Element of REAL , z1 being Element of REAL m such that
A2: ( x = <*q1*> & z1 = 0. (REAL-NS m) & (reproj (i,(0. (REAL-NS m)))) . x = (reproj (i,z1)) . q1 ) by PDIFF_1:def_6;
consider q12 being Element of REAL , z12 being Element of REAL m such that
A3: ( a * x = <*q12*> & z12 = 0. (REAL-NS m) & (reproj (i,(0. (REAL-NS m)))) . (a * x) = (reproj (i,z12)) . q12 ) by PDIFF_1:def_6;
A4: 0. (REAL-NS m) = 0* m by REAL_NS1:def_4;
reconsider qq1 = <*q1*> as Element of REAL 1 by FINSEQ_2:98;
 a * x = a * qq1 by A2, REAL_NS1:3;
then A5: a * x = <*(a * q1)*> by RVSUM_1:47;
a * ((reproj (i,(0. (REAL-NS m)))) . x) = a * ((reproj (i,(0* m))) . q1) by A2, A4, REAL_NS1:3
.= (reproj (i,(0* m))) . (a * q1) by A1, Th15 ;
hence  (reproj (i,(0. (REAL-NS m)))) . (a * x) = a * ((reproj (i,(0. (REAL-NS m)))) . x) by A5, A3, A4, FINSEQ_1:76; ::_thesis: verum
end;

theorem Th17: :: PDIFF_7:17
for m being non empty Element of NAT
for x being Element of REAL
for i being Element of NAT st 1 <= i & i <= m & x <> 0 holds
(reproj (i,(0* m))) . x <> 0* m
proof
let m be non empty Element of NAT ; ::_thesis: for x being Element of REAL
for i being Element of NAT st 1 <= i & i <= m & x <> 0 holds
(reproj (i,(0* m))) . x <> 0* m
let x be Element of REAL ; ::_thesis: for i being Element of NAT st 1 <= i & i <= m & x <> 0 holds
(reproj (i,(0* m))) . x <> 0* m
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m & x <> 0 implies (reproj (i,(0* m))) . x <> 0* m )
assume  ( 1 <= i & i <= m & x <> 0 ) ; ::_thesis: (reproj (i,(0* m))) . x <> 0* m
then  Replace ((0* m),i,x) <> 0* m by Th11;
hence  (reproj (i,(0* m))) . x <> 0* m by PDIFF_1:def_5; ::_thesis: verum
end;

theorem Th18: :: PDIFF_7:18
for m being non empty Element of NAT
for x being Point of (REAL-NS 1)
for i being Element of NAT st 1 <= i & i <= m & x <> 0. (REAL-NS 1) holds
(reproj (i,(0. (REAL-NS m)))) . x <> 0. (REAL-NS m)
proof
let m be non empty Element of NAT ; ::_thesis: for x being Point of (REAL-NS 1)
for i being Element of NAT st 1 <= i & i <= m & x <> 0. (REAL-NS 1) holds
(reproj (i,(0. (REAL-NS m)))) . x <> 0. (REAL-NS m)
let x be Point of (REAL-NS 1); ::_thesis: for i being Element of NAT st 1 <= i & i <= m & x <> 0. (REAL-NS 1) holds
(reproj (i,(0. (REAL-NS m)))) . x <> 0. (REAL-NS m)
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m & x <> 0. (REAL-NS 1) implies (reproj (i,(0. (REAL-NS m)))) . x <> 0. (REAL-NS m) )
assume A1: ( 1 <= i & i <= m & x <> 0. (REAL-NS 1) ) ; ::_thesis: (reproj (i,(0. (REAL-NS m)))) . x <> 0. (REAL-NS m)
consider q1 being Element of REAL , z1 being Element of REAL m such that
A2: ( x = <*q1*> & z1 = 0. (REAL-NS m) & (reproj (i,(0. (REAL-NS m)))) . x = (reproj (i,z1)) . q1 ) by PDIFF_1:def_6;
A3: 0. (REAL-NS m) = 0* m by REAL_NS1:def_4;
now__::_thesis:_not_q1_=_0
assume  q1 = 0 ; ::_thesis: contradiction
then  <*q1*> = 0* 1 by FINSEQ_2:59;
hence  contradiction by A2, A1, REAL_NS1:def_4; ::_thesis: verum
end;
hence  (reproj (i,(0. (REAL-NS m)))) . x <> 0. (REAL-NS m) by A2, A3, A1, Th17; ::_thesis: verum
end;

theorem Th19: :: PDIFF_7:19
for m being non empty Element of NAT
for x, y being Element of REAL
for z being Element of REAL m
for i being Element of NAT st 1 <= i & i <= m & y = (proj (i,m)) . z holds
( ((reproj (i,z)) . x) - z = (reproj (i,(0* m))) . (x - y) & z - ((reproj (i,z)) . x) = (reproj (i,(0* m))) . (y - x) )
proof
let m be non empty Element of NAT ; ::_thesis: for x, y being Element of REAL
for z being Element of REAL m
for i being Element of NAT st 1 <= i & i <= m & y = (proj (i,m)) . z holds
( ((reproj (i,z)) . x) - z = (reproj (i,(0* m))) . (x - y) & z - ((reproj (i,z)) . x) = (reproj (i,(0* m))) . (y - x) )
let x, y be Element of REAL ; ::_thesis: for z being Element of REAL m
for i being Element of NAT st 1 <= i & i <= m & y = (proj (i,m)) . z holds
( ((reproj (i,z)) . x) - z = (reproj (i,(0* m))) . (x - y) & z - ((reproj (i,z)) . x) = (reproj (i,(0* m))) . (y - x) )
let z be Element of REAL m; ::_thesis: for i being Element of NAT st 1 <= i & i <= m & y = (proj (i,m)) . z holds
( ((reproj (i,z)) . x) - z = (reproj (i,(0* m))) . (x - y) & z - ((reproj (i,z)) . x) = (reproj (i,(0* m))) . (y - x) )
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m & y = (proj (i,m)) . z implies ( ((reproj (i,z)) . x) - z = (reproj (i,(0* m))) . (x - y) & z - ((reproj (i,z)) . x) = (reproj (i,(0* m))) . (y - x) ) )
assume  ( 1 <= i & i <= m & y = (proj (i,m)) . z ) ; ::_thesis: ( ((reproj (i,z)) . x) - z = (reproj (i,(0* m))) . (x - y) & z - ((reproj (i,z)) . x) = (reproj (i,(0* m))) . (y - x) )
then A1: ( (Replace (z,i,x)) - z = Replace ((0* m),i,(x - y)) & z - (Replace (z,i,x)) = Replace ((0* m),i,(y - x)) ) by Th12;
 Replace (z,i,x) = (reproj (i,z)) . x by PDIFF_1:def_5;
hence  ( ((reproj (i,z)) . x) - z = (reproj (i,(0* m))) . (x - y) & z - ((reproj (i,z)) . x) = (reproj (i,(0* m))) . (y - x) ) by A1, PDIFF_1:def_5; ::_thesis: verum
end;

theorem Th20: :: PDIFF_7:20
for m being non empty Element of NAT
for x, y being Point of (REAL-NS 1)
for i being Element of NAT
for z being Point of (REAL-NS m) st 1 <= i & i <= m & y = (Proj (i,m)) . z holds
( ((reproj (i,z)) . x) - z = (reproj (i,(0. (REAL-NS m)))) . (x - y) & z - ((reproj (i,z)) . x) = (reproj (i,(0. (REAL-NS m)))) . (y - x) )
proof
let m be non empty Element of NAT ; ::_thesis: for x, y being Point of (REAL-NS 1)
for i being Element of NAT
for z being Point of (REAL-NS m) st 1 <= i & i <= m & y = (Proj (i,m)) . z holds
( ((reproj (i,z)) . x) - z = (reproj (i,(0. (REAL-NS m)))) . (x - y) & z - ((reproj (i,z)) . x) = (reproj (i,(0. (REAL-NS m)))) . (y - x) )
let x, y be Point of (REAL-NS 1); ::_thesis: for i being Element of NAT
for z being Point of (REAL-NS m) st 1 <= i & i <= m & y = (Proj (i,m)) . z holds
( ((reproj (i,z)) . x) - z = (reproj (i,(0. (REAL-NS m)))) . (x - y) & z - ((reproj (i,z)) . x) = (reproj (i,(0. (REAL-NS m)))) . (y - x) )
let i be Element of NAT ; ::_thesis: for z being Point of (REAL-NS m) st 1 <= i & i <= m & y = (Proj (i,m)) . z holds
( ((reproj (i,z)) . x) - z = (reproj (i,(0. (REAL-NS m)))) . (x - y) & z - ((reproj (i,z)) . x) = (reproj (i,(0. (REAL-NS m)))) . (y - x) )
let z be Point of (REAL-NS m); ::_thesis: ( 1 <= i & i <= m & y = (Proj (i,m)) . z implies ( ((reproj (i,z)) . x) - z = (reproj (i,(0. (REAL-NS m)))) . (x - y) & z - ((reproj (i,z)) . x) = (reproj (i,(0. (REAL-NS m)))) . (y - x) ) )
assume A1: ( 1 <= i & i <= m & y = (Proj (i,m)) . z ) ; ::_thesis: ( ((reproj (i,z)) . x) - z = (reproj (i,(0. (REAL-NS m)))) . (x - y) & z - ((reproj (i,z)) . x) = (reproj (i,(0. (REAL-NS m)))) . (y - x) )
consider q1 being Element of REAL , z1 being Element of REAL m such that
A2: ( x = <*q1*> & z1 = z & (reproj (i,z)) . x = (reproj (i,z1)) . q1 ) by PDIFF_1:def_6;
consider q2 being Element of REAL , z2 being Element of REAL m such that
A3: ( y = <*q2*> & z2 = z & (reproj (i,z)) . y = (reproj (i,z2)) . q2 ) by PDIFF_1:def_6;
consider q12 being Element of REAL , z12 being Element of REAL m such that
A4: ( x - y = <*q12*> & z12 = 0. (REAL-NS m) & (reproj (i,(0. (REAL-NS m)))) . (x - y) = (reproj (i,z12)) . q12 ) by PDIFF_1:def_6;
consider q21 being Element of REAL , z21 being Element of REAL m such that
A5: ( y - x = <*q21*> & z21 = 0. (REAL-NS m) & (reproj (i,(0. (REAL-NS m)))) . (y - x) = (reproj (i,z21)) . q21 ) by PDIFF_1:def_6;
A6: 0. (REAL-NS m) = 0* m by REAL_NS1:def_4;
reconsider qq1 = <*q1*> as Element of REAL 1 by FINSEQ_2:98;
reconsider qq2 = <*q2*> as Element of REAL 1 by FINSEQ_2:98;
 ( x - y = qq1 - qq2 & y - x = qq2 - qq1 ) by A2, A3, REAL_NS1:5;
then  ( x - y = <*(q1 - q2)*> & y - x = <*(q2 - q1)*> ) by RVSUM_1:29;
then A7: ( (reproj (i,(0. (REAL-NS m)))) . (x - y) = (reproj (i,(0* m))) . (q1 - q2) & (reproj (i,(0. (REAL-NS m)))) . (y - x) = (reproj (i,(0* m))) . (q2 - q1) ) by A4, A5, A6, FINSEQ_1:76;
 y = <*((proj (i,m)) . z)*> by A1, PDIFF_1:def_4;
then  q2 = (proj (i,m)) . z1 by A2, A3, FINSEQ_1:76;
then  ( ((reproj (i,z1)) . q1) - z1 = (reproj (i,(0* m))) . (q1 - q2) & z1 - ((reproj (i,z1)) . q1) = (reproj (i,(0* m))) . (q2 - q1) ) by Th19, A1;
hence  ( ((reproj (i,z)) . x) - z = (reproj (i,(0. (REAL-NS m)))) . (x - y) & z - ((reproj (i,z)) . x) = (reproj (i,(0. (REAL-NS m)))) . (y - x) ) by A7, A2, REAL_NS1:5; ::_thesis: verum
end;

theorem Th21: :: PDIFF_7:21
for n, m being non empty Element of NAT
for i being Element of NAT
for f being PartFunc of (REAL-NS m),(REAL-NS n)
for x being Point of (REAL-NS m) st f is_differentiable_in x & 1 <= i & i <= m holds
( f is_partial_differentiable_in x,i & partdiff (f,x,i) = (diff (f,x)) * (reproj (i,(0. (REAL-NS m)))) )
proof
let n, m be non empty Element of NAT ; ::_thesis: for i being Element of NAT
for f being PartFunc of (REAL-NS m),(REAL-NS n)
for x being Point of (REAL-NS m) st f is_differentiable_in x & 1 <= i & i <= m holds
( f is_partial_differentiable_in x,i & partdiff (f,x,i) = (diff (f,x)) * (reproj (i,(0. (REAL-NS m)))) )
let i be Element of NAT ; ::_thesis: for f being PartFunc of (REAL-NS m),(REAL-NS n)
for x being Point of (REAL-NS m) st f is_differentiable_in x & 1 <= i & i <= m holds
( f is_partial_differentiable_in x,i & partdiff (f,x,i) = (diff (f,x)) * (reproj (i,(0. (REAL-NS m)))) )
let f be PartFunc of (REAL-NS m),(REAL-NS n); ::_thesis: for x being Point of (REAL-NS m) st f is_differentiable_in x & 1 <= i & i <= m holds
( f is_partial_differentiable_in x,i & partdiff (f,x,i) = (diff (f,x)) * (reproj (i,(0. (REAL-NS m)))) )
let x be Point of (REAL-NS m); ::_thesis: ( f is_differentiable_in x & 1 <= i & i <= m implies ( f is_partial_differentiable_in x,i & partdiff (f,x,i) = (diff (f,x)) * (reproj (i,(0. (REAL-NS m)))) ) )
assume A1: f is_differentiable_in x ; ::_thesis: ( not 1 <= i or not i <= m or ( f is_partial_differentiable_in x,i & partdiff (f,x,i) = (diff (f,x)) * (reproj (i,(0. (REAL-NS m)))) ) )
assume A2: ( 1 <= i & i <= m ) ; ::_thesis: ( f is_partial_differentiable_in x,i & partdiff (f,x,i) = (diff (f,x)) * (reproj (i,(0. (REAL-NS m)))) )
consider N being Neighbourhood of x such that
A3: ( N c= dom f & ex R being RestFunc of (REAL-NS m),(REAL-NS n) st
for y being Point of (REAL-NS m) st y in N holds
(f /. y) - (f /. x) = ((diff (f,x)) . (y - x)) + (R /. (y - x)) ) by A1, NDIFF_1:def_7;
consider R being RestFunc of (REAL-NS m),(REAL-NS n) such that
A4: for y being Point of (REAL-NS m) st y in N holds
(f /. y) - (f /. x) = ((diff (f,x)) . (y - x)) + (R /. (y - x)) by A3;
consider r0 being Real such that
A5: ( 0 < r0 &  {  z where z is Point of (REAL-NS m) : ||.(z - x).|| < r0  }  c= N ) by NFCONT_1:def_1;
set u = f * (reproj (i,x));
reconsider x0 = (Proj (i,m)) . x as Point of (REAL-NS 1) ;
set Z = 0. (REAL-NS m);
set Nx0 =  {  z where z is Point of (REAL-NS 1) : ||.(z - x0).|| < r0  }  ;
now__::_thesis:_for_s_being_set_st_s_in__{__z_where_z_is_Point_of_(REAL-NS_1)_:_||.(z_-_x0).||_<_r0__}__holds_
s_in_the_carrier_of_(REAL-NS_1)
let s be set ; ::_thesis: ( s in  {  z where z is Point of (REAL-NS 1) : ||.(z - x0).|| < r0  }  implies s in the carrier of (REAL-NS 1) )
assume  s in  {  z where z is Point of (REAL-NS 1) : ||.(z - x0).|| < r0  }  ; ::_thesis: s in the carrier of (REAL-NS 1)
then  ex z being Point of (REAL-NS 1) st
( s = z & ||.(z - x0).|| < r0 ) ;
hence  s in the carrier of (REAL-NS 1) ; ::_thesis: verum
end;
then   {  z where z is Point of (REAL-NS 1) : ||.(z - x0).|| < r0  }  is Subset of (REAL-NS 1) by TARSKI:def_3;
then reconsider Nx0 =  {  z where z is Point of (REAL-NS 1) : ||.(z - x0).|| < r0  }  as Neighbourhood of x0 by A5, NFCONT_1:def_1;
A6: for xi being Element of (REAL-NS 1) st xi in Nx0 holds
(reproj (i,x)) . xi in N
proof
let xi be Element of (REAL-NS 1); ::_thesis: ( xi in Nx0 implies (reproj (i,x)) . xi in N )
assume  xi in Nx0 ; ::_thesis: (reproj (i,x)) . xi in N
then A7: ex z being Point of (REAL-NS 1) st
( xi = z & ||.(z - x0).|| < r0 ) ;
 ((reproj (i,x)) . xi) - x = (reproj (i,(0. (REAL-NS m)))) . (xi - x0) by A2, Th20;
then  ||.(((reproj (i,x)) . xi) - x).|| < r0 by A2, Th5, A7;
then  (reproj (i,x)) . xi in  {  z where z is Point of (REAL-NS m) : ||.(z - x).|| < r0  }  ;
hence  (reproj (i,x)) . xi in N by A5; ::_thesis: verum
end;
A8: R is total by NDIFF_1:def_5;
then A9: dom R = the carrier of (REAL-NS m) by PARTFUN1:def_2;
reconsider R1 = R * (reproj (i,(0. (REAL-NS m)))) as PartFunc of (REAL-NS 1),(REAL-NS n) ;
A10: dom (reproj (i,(0. (REAL-NS m)))) = the carrier of (REAL-NS 1) by FUNCT_2:def_1;
A11: dom R1 = the carrier of (REAL-NS 1) by A8, PARTFUN1:def_2;
 for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Point of (REAL-NS 1) st z <> 0. (REAL-NS 1) & ||.z.|| < d holds
(||.z.|| ") * ||.(R1 /. z).|| < r ) )
proof
let r be Real; ::_thesis: ( r > 0 implies ex d being Real st
( d > 0 & ( for z being Point of (REAL-NS 1) st z <> 0. (REAL-NS 1) & ||.z.|| < d holds
(||.z.|| ") * ||.(R1 /. z).|| < r ) ) )
assume  r > 0 ; ::_thesis: ex d being Real st
( d > 0 & ( for z being Point of (REAL-NS 1) st z <> 0. (REAL-NS 1) & ||.z.|| < d holds
(||.z.|| ") * ||.(R1 /. z).|| < r ) )
then consider d being Real such that
A12: ( d > 0 & ( for z being Point of (REAL-NS m) st z <> 0. (REAL-NS m) & ||.z.|| < d holds
(||.z.|| ") * ||.(R /. z).|| < r ) ) by A8, NDIFF_1:23;
take  d ; ::_thesis: ( d > 0 & ( for z being Point of (REAL-NS 1) st z <> 0. (REAL-NS 1) & ||.z.|| < d holds
(||.z.|| ") * ||.(R1 /. z).|| < r ) )
now__::_thesis:_for_z_being_Point_of_(REAL-NS_1)_st_z_<>_0._(REAL-NS_1)_&_||.z.||_<_d_holds_
(||.z.||_")_*_||.(R1_/._z).||_<_r
let z be Point of (REAL-NS 1); ::_thesis: ( z <> 0. (REAL-NS 1) & ||.z.|| < d implies (||.z.|| ") * ||.(R1 /. z).|| < r )
assume A13: ( z <> 0. (REAL-NS 1) & ||.z.|| < d ) ; ::_thesis: (||.z.|| ") * ||.(R1 /. z).|| < r
A14: ||.((reproj (i,(0. (REAL-NS m)))) . z).|| = ||.z.|| by A2, Th5;
 R /. ((reproj (i,(0. (REAL-NS m)))) . z) = R . ((reproj (i,(0. (REAL-NS m)))) . z) by A9, PARTFUN1:def_6;
then  R /. ((reproj (i,(0. (REAL-NS m)))) . z) = R1 . z by A10, FUNCT_1:13;
then  R /. ((reproj (i,(0. (REAL-NS m)))) . z) = R1 /. z by A11, PARTFUN1:def_6;
hence  (||.z.|| ") * ||.(R1 /. z).|| < r by A12, A14, A13, Th18, A2; ::_thesis: verum
end;
hence  ( d > 0 & ( for z being Point of (REAL-NS 1) st z <> 0. (REAL-NS 1) & ||.z.|| < d holds
(||.z.|| ") * ||.(R1 /. z).|| < r ) ) by A12; ::_thesis: verum
end;
then reconsider R1 = R1 as RestFunc of (REAL-NS 1),(REAL-NS n) by A8, NDIFF_1:23;
reconsider dfx = diff (f,x) as Lipschitzian LinearOperator of (REAL-NS m),(REAL-NS n) by LOPBAN_1:def_9;
reconsider LD1 = dfx * (reproj (i,(0. (REAL-NS m)))) as Function of (REAL-NS 1),(REAL-NS n) ;
A15: now__::_thesis:_for_x,_y_being_Element_of_(REAL-NS_1)_holds_LD1_._(x_+_y)_=_(LD1_._x)_+_(LD1_._y)
let x, y be Element of (REAL-NS 1); ::_thesis: LD1 . (x + y) = (LD1 . x) + (LD1 . y)
 LD1 . (x + y) = dfx . ((reproj (i,(0. (REAL-NS m)))) . (x + y)) by FUNCT_2:15;
then  LD1 . (x + y) = dfx . (((reproj (i,(0. (REAL-NS m)))) . x) + ((reproj (i,(0. (REAL-NS m)))) . y)) by Th14, A2;
then  LD1 . (x + y) = (dfx . ((reproj (i,(0. (REAL-NS m)))) . x)) + (dfx . ((reproj (i,(0. (REAL-NS m)))) . y)) by VECTSP_1:def_20;
then  LD1 . (x + y) = (LD1 . x) + (dfx . ((reproj (i,(0. (REAL-NS m)))) . y)) by FUNCT_2:15;
hence  LD1 . (x + y) = (LD1 . x) + (LD1 . y) by FUNCT_2:15; ::_thesis: verum
end;
now__::_thesis:_for_x_being_Element_of_(REAL-NS_1)
for_a_being_Real_holds_LD1_._(a_*_x)_=_a_*_(LD1_._x)
let x be Element of (REAL-NS 1); ::_thesis: for a being Real holds LD1 . (a * x) = a * (LD1 . x)
let a be Real; ::_thesis: LD1 . (a * x) = a * (LD1 . x)
 LD1 . (a * x) = dfx . ((reproj (i,(0. (REAL-NS m)))) . (a * x)) by FUNCT_2:15;
then  LD1 . (a * x) = dfx . (a * ((reproj (i,(0. (REAL-NS m)))) . x)) by Th16, A2;
then  LD1 . (a * x) = a * (dfx . ((reproj (i,(0. (REAL-NS m)))) . x)) by LOPBAN_1:def_5;
hence  LD1 . (a * x) = a * (LD1 . x) by FUNCT_2:15; ::_thesis: verum
end;
then reconsider LD1 = LD1 as LinearOperator of (REAL-NS 1),(REAL-NS n) by A15, LOPBAN_1:def_5, VECTSP_1:def_20;
reconsider LD1 = LD1 as Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))) by LOPBAN_1:def_9;
now__::_thesis:_for_s_being_set_st_s_in_(reproj_(i,x))_.:_Nx0_holds_
s_in_dom_f
let s be set ; ::_thesis: ( s in (reproj (i,x)) .: Nx0 implies s in dom f )
assume  s in (reproj (i,x)) .: Nx0 ; ::_thesis: s in dom f
then  ex t being Element of (REAL-NS 1) st
( t in Nx0 & s = (reproj (i,x)) . t ) by FUNCT_2:65;
then  s in N by A6;
hence  s in dom f by A3; ::_thesis: verum
end;
then A16: (reproj (i,x)) .: Nx0 c= dom f by TARSKI:def_3;
 dom (reproj (i,x)) = the carrier of (REAL-NS 1) by FUNCT_2:def_1;
then A17: Nx0 c= dom (f * (reproj (i,x))) by A16, FUNCT_3:3;
A18: for y being Point of (REAL-NS 1) st y in Nx0 holds
((f * (reproj (i,x))) /. y) - ((f * (reproj (i,x))) /. x0) = (LD1 . (y - x0)) + (R1 /. (y - x0))
proof
let y be Point of (REAL-NS 1); ::_thesis: ( y in Nx0 implies ((f * (reproj (i,x))) /. y) - ((f * (reproj (i,x))) /. x0) = (LD1 . (y - x0)) + (R1 /. (y - x0)) )
assume A19: y in Nx0 ; ::_thesis: ((f * (reproj (i,x))) /. y) - ((f * (reproj (i,x))) /. x0) = (LD1 . (y - x0)) + (R1 /. (y - x0))
then A20: (reproj (i,x)) . y in N by A6;
consider q being Element of REAL , z being Element of REAL m such that
A21: ( x0 = <*q*> & z = x & (reproj (i,x)) . x0 = (reproj (i,z)) . q ) by PDIFF_1:def_6;
reconsider zi = z . i as Element of REAL ;
 x0 = <*((proj (i,m)) . x)*> by PDIFF_1:def_4;
then  q = (proj (i,m)) . z by A21, FINSEQ_1:76;
then  (reproj (i,x)) . x0 = (reproj (i,z)) . (z . i) by A21, PDIFF_1:def_1;
then  (reproj (i,x)) . x0 = Replace (z,i,zi) by PDIFF_1:def_5;
then A22: (reproj (i,x)) . x0 = x by A21, FUNCT_7:35;
A23: x0 in Nx0 by NFCONT_1:4;
A24: (reproj (i,x)) . x0 in N by A6, NFCONT_1:4;
 (f * (reproj (i,x))) /. y = (f * (reproj (i,x))) . y by A19, A17, PARTFUN1:def_6;
then  (f * (reproj (i,x))) /. y = f . ((reproj (i,x)) . y) by FUNCT_2:15;
then A25: (f * (reproj (i,x))) /. y = f /. ((reproj (i,x)) . y) by A20, A3, PARTFUN1:def_6;
 (f * (reproj (i,x))) /. x0 = (f * (reproj (i,x))) . x0 by A23, A17, PARTFUN1:def_6;
then  (f * (reproj (i,x))) /. x0 = f . ((reproj (i,x)) . x0) by FUNCT_2:15;
then A26: ((f * (reproj (i,x))) /. y) - ((f * (reproj (i,x))) /. x0) = (f /. ((reproj (i,x)) . y)) - (f /. x) by A25, A22, A24, A3, PARTFUN1:def_6;
 R /. ((reproj (i,(0. (REAL-NS m)))) . (y - x0)) = R . ((reproj (i,(0. (REAL-NS m)))) . (y - x0)) by A9, PARTFUN1:def_6;
then  R /. ((reproj (i,(0. (REAL-NS m)))) . (y - x0)) = R1 . (y - x0) by A10, FUNCT_1:13;
then A27: R /. ((reproj (i,(0. (REAL-NS m)))) . (y - x0)) = R1 /. (y - x0) by A11, PARTFUN1:def_6;
 ((f * (reproj (i,x))) /. y) - ((f * (reproj (i,x))) /. x0) = ((diff (f,x)) . (((reproj (i,x)) . y) - x)) + (R /. (((reproj (i,x)) . y) - x)) by A26, A4, A19, A6;
then  ((f * (reproj (i,x))) /. y) - ((f * (reproj (i,x))) /. x0) = (dfx . ((reproj (i,(0. (REAL-NS m)))) . (y - x0))) + (R /. (((reproj (i,x)) . y) - x)) by A2, Th20;
then  ((f * (reproj (i,x))) /. y) - ((f * (reproj (i,x))) /. x0) = (dfx . ((reproj (i,(0. (REAL-NS m)))) . (y - x0))) + (R /. ((reproj (i,(0. (REAL-NS m)))) . (y - x0))) by A2, Th20;
hence  ((f * (reproj (i,x))) /. y) - ((f * (reproj (i,x))) /. x0) = (LD1 . (y - x0)) + (R1 /. (y - x0)) by A27, FUNCT_2:15; ::_thesis: verum
end;
then A28: f * (reproj (i,x)) is_differentiable_in x0 by A17, NDIFF_1:def_6;
hence  f is_partial_differentiable_in x,i by PDIFF_1:def_9; ::_thesis: partdiff (f,x,i) = (diff (f,x)) * (reproj (i,(0. (REAL-NS m))))
thus  partdiff (f,x,i) = (diff (f,x)) * (reproj (i,(0. (REAL-NS m)))) by A28, A17, A18, NDIFF_1:def_7; ::_thesis: verum
end;

theorem Th22: :: PDIFF_7:22
for n, m being non empty Element of NAT
for i being Element of NAT
for g being PartFunc of (REAL m),(REAL n)
for y being Element of REAL m st g is_differentiable_in y & 1 <= i & i <= m holds
( g is_partial_differentiable_in y,i & partdiff (g,y,i) = ((diff (g,y)) * (reproj (i,(0. (REAL-NS m))))) . <*1*> )
proof
let n, m be non empty Element of NAT ; ::_thesis: for i being Element of NAT
for g being PartFunc of (REAL m),(REAL n)
for y being Element of REAL m st g is_differentiable_in y & 1 <= i & i <= m holds
( g is_partial_differentiable_in y,i & partdiff (g,y,i) = ((diff (g,y)) * (reproj (i,(0. (REAL-NS m))))) . <*1*> )
let i be Element of NAT ; ::_thesis: for g being PartFunc of (REAL m),(REAL n)
for y being Element of REAL m st g is_differentiable_in y & 1 <= i & i <= m holds
( g is_partial_differentiable_in y,i & partdiff (g,y,i) = ((diff (g,y)) * (reproj (i,(0. (REAL-NS m))))) . <*1*> )
let g be PartFunc of (REAL m),(REAL n); ::_thesis: for y being Element of REAL m st g is_differentiable_in y & 1 <= i & i <= m holds
( g is_partial_differentiable_in y,i & partdiff (g,y,i) = ((diff (g,y)) * (reproj (i,(0. (REAL-NS m))))) . <*1*> )
let y be Element of REAL m; ::_thesis: ( g is_differentiable_in y & 1 <= i & i <= m implies ( g is_partial_differentiable_in y,i & partdiff (g,y,i) = ((diff (g,y)) * (reproj (i,(0. (REAL-NS m))))) . <*1*> ) )
assume A1: ( g is_differentiable_in y & 1 <= i & i <= m ) ; ::_thesis: ( g is_partial_differentiable_in y,i & partdiff (g,y,i) = ((diff (g,y)) * (reproj (i,(0. (REAL-NS m))))) . <*1*> )
then consider f being PartFunc of (REAL-NS m),(REAL-NS n), x being Point of (REAL-NS m) such that
A2: ( f = g & x = y & f is_differentiable_in x ) by PDIFF_1:def_7;
A3: ex f2 being PartFunc of (REAL-NS m),(REAL-NS n) ex x2 being Point of (REAL-NS m) st
( f2 = g & x2 = y & diff (g,y) = diff (f2,x2) ) by A1, PDIFF_1:def_8;
A4: ( f is_partial_differentiable_in x,i & partdiff (f,x,i) = (diff (f,x)) * (reproj (i,(0. (REAL-NS m)))) ) by Th21, A2, A1;
then  g is_partial_differentiable_in y,i by A2, PDIFF_1:def_13;
then  ex f1 being PartFunc of (REAL-NS m),(REAL-NS n) ex x1 being Point of (REAL-NS m) st
( f1 = g & x1 = y & partdiff (g,y,i) = (partdiff (f1,x1,i)) . <*1*> ) by PDIFF_1:def_14;
hence  ( g is_partial_differentiable_in y,i & partdiff (g,y,i) = ((diff (g,y)) * (reproj (i,(0. (REAL-NS m))))) . <*1*> ) by A4, A3, A2, PDIFF_1:def_13; ::_thesis: verum
end;

definition
let n be non empty Element of NAT ;
let f be PartFunc of (REAL n),REAL;
let x be Element of REAL n;
predf is_differentiable_in x means :Def1: :: PDIFF_7:def 1
 <>* f is_differentiable_in x;
end;

:: deftheorem Def1 defines is_differentiable_in PDIFF_7:def_1_:_
 for n being non empty Element of NAT
for f being PartFunc of (REAL n),REAL
for x being Element of REAL n holds
( f is_differentiable_in x iff <>* f is_differentiable_in x );

definition
let n be non empty Element of NAT ;
let f be PartFunc of (REAL n),REAL;
let x be Element of REAL n;
func diff (f,x) ->  Function of (REAL n),REAL equals :: PDIFF_7:def 2
(proj (1,1)) * (diff ((<>* f),x));
coherence
 (proj (1,1)) * (diff ((<>* f),x)) is Function of (REAL n),REAL ;
end;

:: deftheorem  defines diff PDIFF_7:def_2_:_
 for n being non empty Element of NAT
for f being PartFunc of (REAL n),REAL
for x being Element of REAL n holds diff (f,x) = (proj (1,1)) * (diff ((<>* f),x));

theorem :: PDIFF_7:23
for m being non empty Element of NAT
for i being Element of NAT
for h being PartFunc of (REAL m),REAL
for y being Element of REAL m st h is_differentiable_in y & 1 <= i & i <= m holds
( h is_partial_differentiable_in y,i & partdiff (h,y,i) = diff ((h * (reproj (i,y))),((proj (i,m)) . y)) & partdiff (h,y,i) = (diff (h,y)) . ((reproj (i,(0* m))) . 1) )
proof
let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT
for h being PartFunc of (REAL m),REAL
for y being Element of REAL m st h is_differentiable_in y & 1 <= i & i <= m holds
( h is_partial_differentiable_in y,i & partdiff (h,y,i) = diff ((h * (reproj (i,y))),((proj (i,m)) . y)) & partdiff (h,y,i) = (diff (h,y)) . ((reproj (i,(0* m))) . 1) )
let i be Element of NAT ; ::_thesis: for h being PartFunc of (REAL m),REAL
for y being Element of REAL m st h is_differentiable_in y & 1 <= i & i <= m holds
( h is_partial_differentiable_in y,i & partdiff (h,y,i) = diff ((h * (reproj (i,y))),((proj (i,m)) . y)) & partdiff (h,y,i) = (diff (h,y)) . ((reproj (i,(0* m))) . 1) )
let h be PartFunc of (REAL m),REAL; ::_thesis: for y being Element of REAL m st h is_differentiable_in y & 1 <= i & i <= m holds
( h is_partial_differentiable_in y,i & partdiff (h,y,i) = diff ((h * (reproj (i,y))),((proj (i,m)) . y)) & partdiff (h,y,i) = (diff (h,y)) . ((reproj (i,(0* m))) . 1) )
let y be Element of REAL m; ::_thesis: ( h is_differentiable_in y & 1 <= i & i <= m implies ( h is_partial_differentiable_in y,i & partdiff (h,y,i) = diff ((h * (reproj (i,y))),((proj (i,m)) . y)) & partdiff (h,y,i) = (diff (h,y)) . ((reproj (i,(0* m))) . 1) ) )
assume A1: ( h is_differentiable_in y & 1 <= i & i <= m ) ; ::_thesis: ( h is_partial_differentiable_in y,i & partdiff (h,y,i) = diff ((h * (reproj (i,y))),((proj (i,m)) . y)) & partdiff (h,y,i) = (diff (h,y)) . ((reproj (i,(0* m))) . 1) )
then  <>* h is_differentiable_in y by Def1;
then A2: ( <>* h is_partial_differentiable_in y,i & partdiff ((<>* h),y,i) = ((diff ((<>* h),y)) * (reproj (i,(0. (REAL-NS m))))) . <*1*> ) by Th22, A1;
then A3: ex g being PartFunc of (REAL-NS m),(REAL-NS 1) ex x being Point of (REAL-NS m) st
( <>* h = g & x = y & g is_partial_differentiable_in x,i ) by PDIFF_1:def_13;
hence  h is_partial_differentiable_in y,i by PDIFF_1:14; ::_thesis: ( partdiff (h,y,i) = diff ((h * (reproj (i,y))),((proj (i,m)) . y)) & partdiff (h,y,i) = (diff (h,y)) . ((reproj (i,(0* m))) . 1) )
thus  partdiff (h,y,i) = diff ((h * (reproj (i,y))),((proj (i,m)) . y)) ; ::_thesis: partdiff (h,y,i) = (diff (h,y)) . ((reproj (i,(0* m))) . 1)
A4: ex k being PartFunc of (REAL-NS m),(REAL-NS 1) ex z being Point of (REAL-NS m) st
( <>* h = k & y = z & partdiff ((<>* h),y,i) = (partdiff (k,z,i)) . <*1*> ) by A2, PDIFF_1:def_14;
 <*1*> in REAL 1 by FINSEQ_2:98;
then A5: <*1*> in the carrier of (REAL-NS 1) by REAL_NS1:def_4;
 <*(partdiff (h,y,i))*> = partdiff ((<>* h),y,i) by A4, A3, PDIFF_1:15;
then A6: <*(partdiff (h,y,i))*> = (diff ((<>* h),y)) . ((reproj (i,(0. (REAL-NS m)))) . <*1*>) by A2, A5, FUNCT_2:15;
 0. (REAL-NS m) = 0* m by REAL_NS1:def_4;
then  ex q being Element of REAL ex z being Element of REAL m st
( <*1*> = <*q*> & z = 0* m & (reproj (i,(0. (REAL-NS m)))) . <*1*> = (reproj (i,z)) . q ) by A5, PDIFF_1:def_6;
then  (reproj (i,(0. (REAL-NS m)))) . <*1*> = (reproj (i,(0* m))) . 1 by FINSEQ_1:76;
then  partdiff (h,y,i) = (proj (1,1)) . ((diff ((<>* h),y)) . ((reproj (i,(0* m))) . 1)) by A6, PDIFF_1:1;
hence  partdiff (h,y,i) = (diff (h,y)) . ((reproj (i,(0* m))) . 1) by FUNCT_2:15; ::_thesis: verum
end;

theorem Th24: :: PDIFF_7:24
for m being non empty Element of NAT
for v, w, u being FinSequence of REAL m st dom v = dom w & u = v + w holds
Sum u = (Sum v) + (Sum w)
proof
let m be non empty Element of NAT ; ::_thesis: for v, w, u being FinSequence of REAL m st dom v = dom w & u = v + w holds
Sum u = (Sum v) + (Sum w)
defpred S1[ Nat] means  for xseq, yseq, zseq being FinSequence of REAL m st $1 = len zseq & len zseq = len xseq & len zseq = len yseq & ( for i being Element of NAT st i in dom zseq holds
zseq /. i = (xseq /. i) + (yseq /. i) ) holds
Sum zseq = (Sum xseq) + (Sum yseq);
A1: S1[ 0 ]
proof
let xseq, yseq, zseq be FinSequence of REAL m; ::_thesis: ( 0 = len zseq & len zseq = len xseq & len zseq = len yseq & ( for i being Element of NAT st i in dom zseq holds
zseq /. i = (xseq /. i) + (yseq /. i) ) implies Sum zseq = (Sum xseq) + (Sum yseq) )
assume A2: ( 0 = len zseq & len zseq = len xseq & len zseq = len yseq & ( for i being Element of NAT st i in dom zseq holds
zseq /. i = (xseq /. i) + (yseq /. i) ) ) ; ::_thesis: Sum zseq = (Sum xseq) + (Sum yseq)
then A3: ( Sum zseq = 0* m & Sum yseq = 0* m ) by EUCLID_7:def_11;
 0* m = 0. (TOP-REAL m) by EUCLID:70;
then (Sum xseq) + (Sum yseq) = (0. (TOP-REAL m)) + (0. (TOP-REAL m)) by A2, A3, EUCLID_7:def_11
.= 0. (TOP-REAL m) by EUCLID:27 ;
hence  Sum zseq = (Sum xseq) + (Sum yseq) by A3, EUCLID:70; ::_thesis: verum
end;
A4: now__::_thesis:_for_i_being_Element_of_NAT_st_S1[i]_holds_
S1[i_+_1]
let i be Element of NAT ; ::_thesis: ( S1[i] implies S1[i + 1] )
assume A5: S1[i] ; ::_thesis: S1[i + 1]
now__::_thesis:_for_xseq,_yseq,_zseq_being_FinSequence_of_REAL_m_st_i_+_1_=_len_zseq_&_len_zseq_=_len_xseq_&_len_zseq_=_len_yseq_&_(_for_k_being_Element_of_NAT_st_k_in_dom_zseq_holds_
zseq_/._k_=_(xseq_/._k)_+_(yseq_/._k)_)_holds_
Sum_zseq_=_(Sum_xseq)_+_(Sum_yseq)
let xseq, yseq, zseq be FinSequence of REAL m; ::_thesis: ( i + 1 = len zseq & len zseq = len xseq & len zseq = len yseq & ( for k being Element of NAT st k in dom zseq holds
zseq /. k = (xseq /. k) + (yseq /. k) ) implies Sum zseq = (Sum xseq) + (Sum yseq) )
assume A6: ( i + 1 = len zseq & len zseq = len xseq & len zseq = len yseq & ( for k being Element of NAT st k in dom zseq holds
zseq /. k = (xseq /. k) + (yseq /. k) ) ) ; ::_thesis: Sum zseq = (Sum xseq) + (Sum yseq)
set xseq0 = xseq | i;
set yseq0 = yseq | i;
set zseq0 = zseq | i;
A7: ( dom xseq = dom yseq & dom zseq = dom yseq ) by A6, FINSEQ_3:29;
A8: i = len (xseq | i) by A6, FINSEQ_1:59, NAT_1:11;
then A9: ( len (xseq | i) = len (yseq | i) & len (xseq | i) = len (zseq | i) ) by A6, FINSEQ_1:59, NAT_1:11;
 for k being Element of NAT st k in dom (zseq | i) holds
(zseq | i) /. k = ((xseq | i) /. k) + ((yseq | i) /. k)
proof
let k be Element of NAT ; ::_thesis: ( k in dom (zseq | i) implies (zseq | i) /. k = ((xseq | i) /. k) + ((yseq | i) /. k) )
assume A10: k in dom (zseq | i) ; ::_thesis: (zseq | i) /. k = ((xseq | i) /. k) + ((yseq | i) /. k)
then A11: ( k in dom (yseq | (Seg i)) & k in dom (xseq | (Seg i)) & k in dom (zseq | (Seg i)) ) by A9, FINSEQ_3:29;
A12: ( k in Seg i & k in dom zseq ) by A10, RELAT_1:57;
then A13: zseq /. k = (xseq /. k) + (yseq /. k) by A6;
A14: xseq /. k = xseq . k by A12, A7, PARTFUN1:def_6
.= (xseq | (Seg i)) . k by A12, FUNCT_1:49
.= (xseq | (Seg i)) /. k by A11, PARTFUN1:def_6 ;
A15: yseq /. k = yseq . k by A7, A12, PARTFUN1:def_6
.= (yseq | (Seg i)) . k by A12, FUNCT_1:49
.= (yseq | (Seg i)) /. k by A11, PARTFUN1:def_6 ;
(zseq | i) /. k = (zseq | (Seg i)) . k by A10, PARTFUN1:def_6
.= zseq . k by A12, FUNCT_1:49 ;
hence  (zseq | i) /. k = ((xseq | i) /. k) + ((yseq | i) /. k) by A13, A14, A15, A12, PARTFUN1:def_6; ::_thesis: verum
end;
then A16: Sum (zseq | i) = (Sum (xseq | i)) + (Sum (yseq | i)) by A8, A9, A5;
consider v being Element of REAL m such that
A17: ( v = xseq . (len xseq) & Sum xseq = (Sum (xseq | i)) + v ) by A6, A8, PDIFF_6:15;
consider w being Element of REAL m such that
A18: ( w = yseq . (len yseq) & Sum yseq = (Sum (yseq | i)) + w ) by A6, A8, A9, PDIFF_6:15;
consider t being Element of REAL m such that
A19: ( t = zseq . (len zseq) & Sum zseq = (Sum (zseq | i)) + t ) by A6, A8, A9, PDIFF_6:15;
A20: dom zseq = Seg (i + 1) by A6, FINSEQ_1:def_3;
then  len zseq in dom zseq by A6, FINSEQ_1:4;
then  t = zseq /. (len zseq) by A19, PARTFUN1:def_6;
then A21: t = (xseq /. (len xseq)) + (yseq /. (len yseq)) by A6, A20, FINSEQ_1:4;
 ( dom xseq = Seg (i + 1) & dom yseq = Seg (i + 1) ) by A6, FINSEQ_1:def_3;
then A22: ( v = xseq /. (len xseq) & w = yseq /. (len yseq) ) by A6, A17, A18, FINSEQ_1:4, PARTFUN1:def_6;
reconsider F1 = Sum (xseq | i) as real-valued FinSequence ;
reconsider F2 = Sum (yseq | i) as real-valued FinSequence ;
reconsider F3 = xseq /. (len xseq) as real-valued FinSequence ;
reconsider F4 = yseq /. (len yseq) as real-valued FinSequence ;
 Sum zseq = ((F1 + F2) + F3) + F4 by A19, A16, A21, RVSUM_1:15;
then  Sum zseq = ((F1 + F3) + F2) + F4 by RVSUM_1:15;
hence  Sum zseq = (Sum xseq) + (Sum yseq) by A22, A17, A18, RVSUM_1:15; ::_thesis: verum
end;
hence  S1[i + 1] ; ::_thesis: verum
end;
A23: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A1, A4);
let xseq, yseq, zseq be FinSequence of REAL m; ::_thesis: ( dom xseq = dom yseq & zseq = xseq + yseq implies Sum zseq = (Sum xseq) + (Sum yseq) )
assume A24: ( dom xseq = dom yseq & zseq = xseq + yseq ) ; ::_thesis: Sum zseq = (Sum xseq) + (Sum yseq)
then A25: len yseq = len xseq by FINSEQ_3:29;
 xseq + yseq = xseq <++> yseq by INTEGR15:def_9;
then  dom zseq = (dom xseq) /\ (dom yseq) by A24, VALUED_2:def_45;
then A26: len zseq = len xseq by A24, FINSEQ_3:29;
 for i being Element of NAT st i in dom zseq holds
zseq /. i = (xseq /. i) + (yseq /. i) by A24, INTEGR15:21;
hence  Sum zseq = (Sum xseq) + (Sum yseq) by A25, A26, A23; ::_thesis: verum
end;

theorem Th25: :: PDIFF_7:25
for m being non empty Element of NAT
for r being Real
for w, u being FinSequence of REAL m st u = r (#) w holds
Sum u = r * (Sum w)
proof
let m be non empty Element of NAT ; ::_thesis: for r being Real
for w, u being FinSequence of REAL m st u = r (#) w holds
Sum u = r * (Sum w)
let r be Real; ::_thesis: for w, u being FinSequence of REAL m st u = r (#) w holds
Sum u = r * (Sum w)
defpred S1[ Nat] means  for xseq, yseq being FinSequence of REAL m st $1 = len xseq & len xseq = len yseq & ( for i being Element of NAT st i in dom xseq holds
yseq /. i = r * (xseq /. i) ) holds
Sum yseq = r * (Sum xseq);
A1: S1[ 0 ]
proof
let xseq, yseq be FinSequence of REAL m; ::_thesis: ( 0 = len xseq & len xseq = len yseq & ( for i being Element of NAT st i in dom xseq holds
yseq /. i = r * (xseq /. i) ) implies Sum yseq = r * (Sum xseq) )
assume A2: ( 0 = len xseq & len xseq = len yseq & ( for i being Element of NAT st i in dom xseq holds
yseq /. i = r * (xseq /. i) ) ) ; ::_thesis: Sum yseq = r * (Sum xseq)
reconsider r1 = r as real number ;
 Sum xseq = 0* m by A2, EUCLID_7:def_11;
then r * (Sum xseq) = r1 * (0. (TOP-REAL m)) by EUCLID:70
.= 0. (TOP-REAL m) by EUCLID:28
.= 0* m by EUCLID:70 ;
hence  Sum yseq = r * (Sum xseq) by A2, EUCLID_7:def_11; ::_thesis: verum
end;
A3: now__::_thesis:_for_i_being_Element_of_NAT_st_S1[i]_holds_
S1[i_+_1]
let i be Element of NAT ; ::_thesis: ( S1[i] implies S1[i + 1] )
assume A4: S1[i] ; ::_thesis: S1[i + 1]
now__::_thesis:_for_xseq,_yseq_being_FinSequence_of_REAL_m_st_i_+_1_=_len_xseq_&_len_xseq_=_len_yseq_&_(_for_k_being_Element_of_NAT_st_k_in_dom_xseq_holds_
yseq_/._k_=_r_*_(xseq_/._k)_)_holds_
Sum_yseq_=_r_*_(Sum_xseq)
let xseq, yseq be FinSequence of REAL m; ::_thesis: ( i + 1 = len xseq & len xseq = len yseq & ( for k being Element of NAT st k in dom xseq holds
yseq /. k = r * (xseq /. k) ) implies Sum yseq = r * (Sum xseq) )
assume A5: ( i + 1 = len xseq & len xseq = len yseq & ( for k being Element of NAT st k in dom xseq holds
yseq /. k = r * (xseq /. k) ) ) ; ::_thesis: Sum yseq = r * (Sum xseq)
then A6: dom xseq = dom yseq by FINSEQ_3:29;
set xseq0 = xseq | i;
set yseq0 = yseq | i;
A7: i = len (xseq | i) by A5, FINSEQ_1:59, NAT_1:11;
then A8: len (xseq | i) = len (yseq | i) by A5, FINSEQ_1:59, NAT_1:11;
 for k being Element of NAT st k in dom (xseq | i) holds
(yseq | i) /. k = r * ((xseq | i) /. k)
proof
let k be Element of NAT ; ::_thesis: ( k in dom (xseq | i) implies (yseq | i) /. k = r * ((xseq | i) /. k) )
assume A9: k in dom (xseq | i) ; ::_thesis: (yseq | i) /. k = r * ((xseq | i) /. k)
then A10: ( k in dom xseq & k in Seg i ) by RELAT_1:57;
A11: k in dom (yseq | (Seg i)) by A9, A8, FINSEQ_3:29;
A12: xseq /. k = xseq . k by A10, PARTFUN1:def_6
.= (xseq | (Seg i)) . k by A10, FUNCT_1:49
.= (xseq | i) /. k by A9, PARTFUN1:def_6 ;
(yseq | i) /. k = (yseq | (Seg i)) . k by A11, PARTFUN1:def_6
.= yseq . k by A10, FUNCT_1:49
.= yseq /. k by A10, A6, PARTFUN1:def_6 ;
hence  (yseq | i) /. k = r * ((xseq | i) /. k) by A5, A10, A12; ::_thesis: verum
end;
then A13: Sum (yseq | i) = r * (Sum (xseq | i)) by A7, A8, A4;
consider v being Element of REAL m such that
A14: ( v = xseq . (len xseq) & Sum xseq = (Sum (xseq | i)) + v ) by A5, A7, PDIFF_6:15;
consider w being Element of REAL m such that
A15: ( w = yseq . (len yseq) & Sum yseq = (Sum (yseq | i)) + w ) by A5, A7, A8, PDIFF_6:15;
A16: dom xseq = Seg (i + 1) by A5, FINSEQ_1:def_3;
then A17: len yseq in dom xseq by A5, FINSEQ_1:4;
then w = yseq /. (len yseq) by A15, A6, PARTFUN1:def_6
.= r * (xseq /. (len yseq)) by A5, A16, FINSEQ_1:4
.= r * v by A17, A5, A14, PARTFUN1:def_6 ;
hence  Sum yseq = r * (Sum xseq) by A15, A13, A14, RVSUM_1:51; ::_thesis: verum
end;
hence  S1[i + 1] ; ::_thesis: verum
end;
A18: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A1, A3);
let xseq, yseq be FinSequence of REAL m; ::_thesis: ( yseq = r (#) xseq implies Sum yseq = r * (Sum xseq) )
A19: r (#) xseq = xseq [#] r by INTEGR15:def_11;
assume A20: yseq = r (#) xseq ; ::_thesis: Sum yseq = r * (Sum xseq)
then A21: dom yseq = dom xseq by A19, VALUED_2:def_39;
then A22: len xseq = len yseq by FINSEQ_3:29;
 for i being Element of NAT st i in dom xseq holds
yseq /. i = r * (xseq /. i) by A20, A21, INTEGR15:23;
hence  Sum yseq = r * (Sum xseq) by A22, A18; ::_thesis: verum
end;

Lm8:  for m, n being non empty Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for x being Element of REAL m ex L being Lipschitzian LinearOperator of m,n st
for h being Element of REAL m ex w being FinSequence of REAL n st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = ((proj (i,m)) . h) * (partdiff (f,x,i)) ) & L . h = Sum w )
proof
let m, n be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n)
for x being Element of REAL m ex L being Lipschitzian LinearOperator of m,n st
for h being Element of REAL m ex w being FinSequence of REAL n st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = ((proj (i,m)) . h) * (partdiff (f,x,i)) ) & L . h = Sum w )
let f be PartFunc of (REAL m),(REAL n); ::_thesis: for x being Element of REAL m ex L being Lipschitzian LinearOperator of m,n st
for h being Element of REAL m ex w being FinSequence of REAL n st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = ((proj (i,m)) . h) * (partdiff (f,x,i)) ) & L . h = Sum w )
let x be Element of REAL m; ::_thesis: ex L being Lipschitzian LinearOperator of m,n st
for h being Element of REAL m ex w being FinSequence of REAL n st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = ((proj (i,m)) . h) * (partdiff (f,x,i)) ) & L . h = Sum w )
defpred S1[ set , set ] means  ex w being FinSequence of REAL n st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = ((proj (i,m)) . $1) * (partdiff (f,x,i)) ) & $2 = Sum w );
A1: for v being Element of REAL m ex y being Element of REAL n st S1[v,y]
proof
let v be Element of REAL m; ::_thesis: ex y being Element of REAL n st S1[v,y]
defpred S2[ set , set ] means  ex i being Element of NAT st
( i = $1 & $2 = ((proj (i,m)) . v) * (partdiff (f,x,i)) );
A2: for i being Nat st i in Seg m holds
ex r being Element of REAL n st S2[i,r]
proof
let i be Nat; ::_thesis: ( i in Seg m implies ex r being Element of REAL n st S2[i,r] )
assume  i in Seg m ; ::_thesis: ex r being Element of REAL n st S2[i,r]
reconsider i0 = i as Element of NAT by ORDINAL1:def_12;
 ((proj (i0,m)) . v) * (partdiff (f,x,i0)) in REAL n ;
hence  ex r being Element of REAL n st S2[i,r] ; ::_thesis: verum
end;
consider w being FinSequence of REAL n such that
A3: ( dom w = Seg m & ( for i being Nat st i in Seg m holds
S2[i,w . i] ) ) from FINSEQ_1:sch_5(A2);
A4: now__::_thesis:_for_i_being_Element_of_NAT_st_i_in_Seg_m_holds_
w_._i_=_((proj_(i,m))_._v)_*_(partdiff_(f,x,i))
let i be Element of NAT ; ::_thesis: ( i in Seg m implies w . i = ((proj (i,m)) . v) * (partdiff (f,x,i)) )
assume  i in Seg m ; ::_thesis: w . i = ((proj (i,m)) . v) * (partdiff (f,x,i))
then  S2[i,w . i] by A3;
hence  w . i = ((proj (i,m)) . v) * (partdiff (f,x,i)) ; ::_thesis: verum
end;
reconsider w0 = Sum w as Element of REAL n ;
 ex w being FinSequence of REAL n st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = ((proj (i,m)) . v) * (partdiff (f,x,i)) ) & w0 = Sum w ) by A4, A3;
hence  ex y0 being Element of REAL n st S1[v,y0] ; ::_thesis: verum
end;
consider L being Function of (REAL m),(REAL n) such that
A5: for h being Element of REAL m holds S1[h,L . h] from FUNCT_2:sch_3(A1);
A6: for s, t being Element of REAL m holds L . (s + t) = (L . s) + (L . t)
proof
let s, t be Element of REAL m; ::_thesis: L . (s + t) = (L . s) + (L . t)
consider w being FinSequence of REAL n such that
A7: ( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = ((proj (i,m)) . s) * (partdiff (f,x,i)) ) & L . s = Sum w ) by A5;
consider v being FinSequence of REAL n such that
A8: ( dom v = Seg m & ( for i being Element of NAT st i in Seg m holds
v . i = ((proj (i,m)) . t) * (partdiff (f,x,i)) ) & L . t = Sum v ) by A5;
consider u being FinSequence of REAL n such that
A9: ( dom u = Seg m & ( for i being Element of NAT st i in Seg m holds
u . i = ((proj (i,m)) . (s + t)) * (partdiff (f,x,i)) ) & L . (s + t) = Sum u ) by A5;
A10: w + v = w <++> v by INTEGR15:def_9;
A11: dom u = (dom w) /\ (dom v) by A7, A8, A9;
now__::_thesis:_for_j_being_set_st_j_in_dom_u_holds_
u_._j_=_(w_._j)_+_(v_._j)
let j be set ; ::_thesis: ( j in dom u implies u . j = (w . j) + (v . j) )
assume A12: j in dom u ; ::_thesis: u . j = (w . j) + (v . j)
then reconsider i = j as Element of NAT ;
A13: w . i = ((proj (i,m)) . s) * (partdiff (f,x,i)) by A7, A9, A12;
A14: v . i = ((proj (i,m)) . t) * (partdiff (f,x,i)) by A8, A9, A12;
thus u . j = ((proj (i,m)) . (s + t)) * (partdiff (f,x,i)) by A9, A12
.= ((s + t) . i) * (partdiff (f,x,i)) by PDIFF_1:def_1
.= ((s . i) + (t . i)) * (partdiff (f,x,i)) by RVSUM_1:11
.= (((proj (i,m)) . s) + (t . i)) * (partdiff (f,x,i)) by PDIFF_1:def_1
.= (((proj (i,m)) . s) + ((proj (i,m)) . t)) * (partdiff (f,x,i)) by PDIFF_1:def_1
.= (w . j) + (v . j) by A13, A14, RVSUM_1:50 ; ::_thesis: verum
end;
then  u = w + v by A10, A11, VALUED_2:def_45;
hence  L . (s + t) = (L . s) + (L . t) by A9, A7, A8, Th24; ::_thesis: verum
end;
 for s being Element of REAL m
for r being Real holds L . (r * s) = r * (L . s)
proof
let s be Element of REAL m; ::_thesis: for r being Real holds L . (r * s) = r * (L . s)
let r be Real; ::_thesis: L . (r * s) = r * (L . s)
consider w being FinSequence of REAL n such that
A15: ( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = ((proj (i,m)) . s) * (partdiff (f,x,i)) ) & L . s = Sum w ) by A5;
consider u being FinSequence of REAL n such that
A16: ( dom u = Seg m & ( for i being Element of NAT st i in Seg m holds
u . i = ((proj (i,m)) . (r * s)) * (partdiff (f,x,i)) ) & L . (r * s) = Sum u ) by A5;
A17: r (#) w = w [#] r by INTEGR15:def_11;
now__::_thesis:_for_j_being_set_st_j_in_dom_u_holds_
u_._j_=_r_(#)_(w_._j)
let j be set ; ::_thesis: ( j in dom u implies u . j = r (#) (w . j) )
assume A18: j in dom u ; ::_thesis: u . j = r (#) (w . j)
then reconsider i = j as Element of NAT ;
A19: w . i = ((proj (i,m)) . s) * (partdiff (f,x,i)) by A15, A16, A18;
thus u . j = ((proj (i,m)) . (r * s)) * (partdiff (f,x,i)) by A16, A18
.= ((r * s) . i) * (partdiff (f,x,i)) by PDIFF_1:def_1
.= (r * (s . i)) * (partdiff (f,x,i)) by RVSUM_1:45
.= (r * ((proj (i,m)) . s)) * (partdiff (f,x,i)) by PDIFF_1:def_1
.= r (#) (w . j) by A19, RVSUM_1:49 ; ::_thesis: verum
end;
then  u = r (#) w by A17, A15, A16, VALUED_2:def_39;
hence  L . (r * s) = r * (L . s) by A15, A16, Th25; ::_thesis: verum
end;
then reconsider L = L as LinearOperator of m,n by A6, PDIFF_6:def_1, PDIFF_6:def_2;
take  L ; ::_thesis: for h being Element of REAL m ex w being FinSequence of REAL n st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = ((proj (i,m)) . h) * (partdiff (f,x,i)) ) & L . h = Sum w )
thus  for h being Element of REAL m ex w being FinSequence of REAL n st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = ((proj (i,m)) . h) * (partdiff (f,x,i)) ) & L . h = Sum w ) by A5; ::_thesis: verum
end;

theorem Th26: :: PDIFF_7:26
for n being non empty Element of NAT
for h, g being FinSequence of REAL n st len h = (len g) + 1 & ( for i being Nat st i in dom g holds
g /. i = (h /. i) - (h /. (i + 1)) ) holds
(h /. 1) - (h /. (len h)) = Sum g
proof
let n be non empty Element of NAT ; ::_thesis: for h, g being FinSequence of REAL n st len h = (len g) + 1 & ( for i being Nat st i in dom g holds
g /. i = (h /. i) - (h /. (i + 1)) ) holds
(h /. 1) - (h /. (len h)) = Sum g
let h, g be FinSequence of REAL n; ::_thesis: ( len h = (len g) + 1 & ( for i being Nat st i in dom g holds
g /. i = (h /. i) - (h /. (i + 1)) ) implies (h /. 1) - (h /. (len h)) = Sum g )
assume that
A1: len h = (len g) + 1 and
A2: for i being Nat st i in dom g holds
g /. i = (h /. i) - (h /. (i + 1)) ; ::_thesis: (h /. 1) - (h /. (len h)) = Sum g
percases ( len g = 0 or len g > 0 ) ;
supposeA3: len g = 0 ; ::_thesis: (h /. 1) - (h /. (len h)) = Sum g
then  (h /. 1) - (h /. (len h)) = 0* n by A1, EUCLIDLP:9;
hence  (h /. 1) - (h /. (len h)) = Sum g by A3, EUCLID_7:def_11; ::_thesis: verum
end;
supposeA4: len g > 0 ; ::_thesis: (h /. 1) - (h /. (len h)) = Sum g
then A5: Sum g = (accum g) . (len g) by EUCLID_7:def_11;
defpred S1[ Nat] means ( $1 <= len g implies (accum g) . $1 = (h /. 1) - (h /. ($1 + 1)) );
A6: S1[1]
proof
assume  1 <= len g ; ::_thesis: (accum g) . 1 = (h /. 1) - (h /. (1 + 1))
then  1 in Seg (len g) ;
then A7: 1 in dom g by FINSEQ_1:def_3;
 (accum g) . 1 = g . 1 by EUCLID_7:def_10;
then  (accum g) . 1 = g /. 1 by A7, PARTFUN1:def_6;
hence  (accum g) . 1 = (h /. 1) - (h /. (1 + 1)) by A7, A2; ::_thesis: verum
end;
A8: for j being Nat st 1 <= j & S1[j] holds
S1[j + 1]
proof
let j be Nat; ::_thesis: ( 1 <= j & S1[j] implies S1[j + 1] )
assume A9: 1 <= j ; ::_thesis: ( not S1[j] or S1[j + 1] )
assume A10: S1[j] ; ::_thesis: S1[j + 1]
assume A11: j + 1 <= len g ; ::_thesis: (accum g) . (j + 1) = (h /. 1) - (h /. ((j + 1) + 1))
then A12: j < len g by NAT_1:13;
 1 <= j + 1 by XREAL_1:38;
then A13: j + 1 in dom g by A11, FINSEQ_3:25;
 len g = len (accum g) by EUCLID_7:def_10;
then A14: j in dom (accum g) by A9, A12, FINSEQ_3:25;
 (accum g) . (j + 1) = ((accum g) /. j) + (g /. (j + 1)) by A9, A12, EUCLID_7:def_10;
then A15: (accum g) . (j + 1) = ((accum g) /. j) + ((h /. (j + 1)) - (h /. ((j + 1) + 1))) by A2, A13;
reconsider hj1 = h /. (j + 1) as Point of (TOP-REAL n) by EUCLID:22;
reconsider hj2 = h /. (j + 2) as Point of (TOP-REAL n) by EUCLID:22;
reconsider hj3 = (h /. 1) - (h /. (j + 1)) as Point of (TOP-REAL n) by EUCLID:22;
 (accum g) . (j + 1) = hj3 + (hj1 - hj2) by A15, A10, A11, A14, NAT_1:13, PARTFUN1:def_6;
then  (accum g) . (j + 1) = (hj3 + hj1) - hj2 by EUCLID:45;
hence  (accum g) . (j + 1) = (h /. 1) - (h /. ((j + 1) + 1)) by RVSUM_1:43; ::_thesis: verum
end;
A16: 1 <= len g by A4, NAT_1:14;
 for i being Nat st 1 <= i holds
S1[i] from NAT_1:sch_8(A6, A8);
hence  (h /. 1) - (h /. (len h)) = Sum g by A5, A1, A16; ::_thesis: verum
end;
end;
end;

theorem Th27: :: PDIFF_7:27
for n being non empty Element of NAT
for h, g, j being FinSequence of REAL n st len h = len j & len g = len j & ( for i being Nat st i in dom j holds
j /. i = (h /. i) - (g /. i) ) holds
Sum j = (Sum h) - (Sum g)
proof
let n be non empty Element of NAT ; ::_thesis: for h, g, j being FinSequence of REAL n st len h = len j & len g = len j & ( for i being Nat st i in dom j holds
j /. i = (h /. i) - (g /. i) ) holds
Sum j = (Sum h) - (Sum g)
let h, g, j be FinSequence of REAL n; ::_thesis: ( len h = len j & len g = len j & ( for i being Nat st i in dom j holds
j /. i = (h /. i) - (g /. i) ) implies Sum j = (Sum h) - (Sum g) )
assume that
A1: ( len h = len j & len g = len j ) and
A2: for i being Nat st i in dom j holds
j /. i = (h /. i) - (g /. i) ; ::_thesis: Sum j = (Sum h) - (Sum g)
A3: ( dom j = Seg (len j) & dom g = Seg (len g) & dom h = Seg (len h) ) by FINSEQ_1:def_3;
A4: for i being Nat st i in dom h holds
h /. i = (j /. i) + (g /. i)
proof
let i be Nat; ::_thesis: ( i in dom h implies h /. i = (j /. i) + (g /. i) )
reconsider ji = j /. i, hi = h /. i, gi = g /. i as Point of (TOP-REAL n) by EUCLID:22;
assume  i in dom h ; ::_thesis: h /. i = (j /. i) + (g /. i)
then  ji = hi - gi by A1, A2, A3;
then  ji + gi = hi by EUCLID:48;
hence  h /. i = (j /. i) + (g /. i) ; ::_thesis: verum
end;
 j + g = j <++> g by INTEGR15:def_9;
then A5: dom (j + g) = (dom j) /\ (dom g) by VALUED_2:def_45;
reconsider Sj = Sum j, Sh = Sum h, Sg = Sum g as Point of (TOP-REAL n) by EUCLID:22;
 for k being Element of NAT st k in dom h holds
h . k = (j + g) . k
proof
let k be Element of NAT ; ::_thesis: ( k in dom h implies h . k = (j + g) . k )
assume A6: k in dom h ; ::_thesis: h . k = (j + g) . k
then  h /. k = (j /. k) + (g /. k) by A4;
then A7: h . k = (j /. k) + (g /. k) by A6, PARTFUN1:def_6;
 (j + g) /. k = (j /. k) + (g /. k) by A6, A1, A3, A5, INTEGR15:21;
hence  h . k = (j + g) . k by A7, A6, A1, A3, A5, PARTFUN1:def_6; ::_thesis: verum
end;
then  Sh = Sj + Sg by A1, A3, Th24, A5, PARTFUN1:5;
then  Sh - Sg = Sj by EUCLID:48;
hence  (Sum h) - (Sum g) = Sum j ; ::_thesis: verum
end;

theorem Th28: :: PDIFF_7:28
for m, n being non empty Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for x, y being Element of REAL m ex h being FinSequence of REAL m ex g being FinSequence of REAL n st
( len h = m + 1 & len g = m & ( for i being Nat st i in dom h holds
h /. i = (y | ((m + 1) -' i)) ^ (0* (i -' 1)) ) & ( for i being Nat st i in dom g holds
g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))) ) & ( for i being Nat
for hi being Element of REAL m st i in dom h & h /. i = hi holds
|.hi.| <= |.y.| ) & (f /. (x + y)) - (f /. x) = Sum g )
proof
let m, n be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n)
for x, y being Element of REAL m ex h being FinSequence of REAL m ex g being FinSequence of REAL n st
( len h = m + 1 & len g = m & ( for i being Nat st i in dom h holds
h /. i = (y | ((m + 1) -' i)) ^ (0* (i -' 1)) ) & ( for i being Nat st i in dom g holds
g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))) ) & ( for i being Nat
for hi being Element of REAL m st i in dom h & h /. i = hi holds
|.hi.| <= |.y.| ) & (f /. (x + y)) - (f /. x) = Sum g )
let f be PartFunc of (REAL m),(REAL n); ::_thesis: for x, y being Element of REAL m ex h being FinSequence of REAL m ex g being FinSequence of REAL n st
( len h = m + 1 & len g = m & ( for i being Nat st i in dom h holds
h /. i = (y | ((m + 1) -' i)) ^ (0* (i -' 1)) ) & ( for i being Nat st i in dom g holds
g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))) ) & ( for i being Nat
for hi being Element of REAL m st i in dom h & h /. i = hi holds
|.hi.| <= |.y.| ) & (f /. (x + y)) - (f /. x) = Sum g )
let x, y be Element of REAL m; ::_thesis: ex h being FinSequence of REAL m ex g being FinSequence of REAL n st
( len h = m + 1 & len g = m & ( for i being Nat st i in dom h holds
h /. i = (y | ((m + 1) -' i)) ^ (0* (i -' 1)) ) & ( for i being Nat st i in dom g holds
g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))) ) & ( for i being Nat
for hi being Element of REAL m st i in dom h & h /. i = hi holds
|.hi.| <= |.y.| ) & (f /. (x + y)) - (f /. x) = Sum g )
A1: len y = m by FINSEQ_2:133;
defpred S1[ Nat, set ] means $2 = (y | ((m + 1) -' $1)) ^ (0* ($1 -' 1));
A2: for k being Nat st k in Seg (m + 1) holds
ex x being Element of REAL m st S1[k,x]
proof
let k be Nat; ::_thesis: ( k in Seg (m + 1) implies ex x being Element of REAL m st S1[k,x] )
assume  k in Seg (m + 1) ; ::_thesis: ex x being Element of REAL m st S1[k,x]
then A3: ( 1 <= k & k <= m + 1 ) by FINSEQ_1:1;
then  ( k - 1 >= 0 & (m + 1) - k >= 0 ) by XREAL_1:48;
then A4: ( (m + 1) -' k = (m + 1) - k & k -' 1 = k - 1 ) by XREAL_0:def_2;
set l1 = (m + 1) -' k;
set l2 = k -' 1;
 m + 1 <= m + k by A3, XREAL_1:6;
then  (m + 1) -' k <= len y by A1, A4, XREAL_1:20;
then A5: len (y | ((m + 1) -' k)) = (m + 1) -' k by FINSEQ_1:59;
 len (0* (k -' 1)) = k -' 1 by FINSEQ_2:132;
then  len ((y | ((m + 1) -' k)) ^ (0* (k -' 1))) = ((m + 1) -' k) + (k -' 1) by A5, FINSEQ_1:22;
then  (y | ((m + 1) -' k)) ^ (0* (k -' 1)) is Element of (((m + 1) -' k) + (k -' 1)) -tuples_on REAL by FINSEQ_2:133;
hence  ex x being Element of REAL m st S1[k,x] by A4; ::_thesis: verum
end;
consider h being FinSequence of REAL m such that
A6: ( dom h = Seg (m + 1) & ( for j being Nat st j in Seg (m + 1) holds
S1[j,h . j] ) ) from FINSEQ_1:sch_5(A2);
A7: now__::_thesis:_for_j_being_Nat_st_j_in_dom_h_holds_
h_/._j_=_(y_|_((m_+_1)_-'_j))_^_(0*_(j_-'_1))
let j be Nat; ::_thesis: ( j in dom h implies h /. j = (y | ((m + 1) -' j)) ^ (0* (j -' 1)) )
assume A8: j in dom h ; ::_thesis: h /. j = (y | ((m + 1) -' j)) ^ (0* (j -' 1))
then  h /. j = h . j by PARTFUN1:def_6;
hence  h /. j = (y | ((m + 1) -' j)) ^ (0* (j -' 1)) by A8, A6; ::_thesis: verum
end;
deffunc H1( Nat) ->  Element of REAL n = f /. (x + (h /. $1));
consider z being FinSequence of REAL n such that
A9: ( len z = m + 1 & ( for j being Nat st j in dom z holds
z . j = H1(j) ) ) from FINSEQ_2:sch_1();
A10: now__::_thesis:_for_j_being_Nat_st_j_in_dom_z_holds_
z_/._j_=_f_/._(x_+_(h_/._j))
let j be Nat; ::_thesis: ( j in dom z implies z /. j = f /. (x + (h /. j)) )
assume A11: j in dom z ; ::_thesis: z /. j = f /. (x + (h /. j))
then  z /. j = z . j by PARTFUN1:def_6;
hence  z /. j = f /. (x + (h /. j)) by A11, A9; ::_thesis: verum
end;
deffunc H2( Nat) ->  Element of REAL n = (z /. $1) - (z /. ($1 + 1));
consider g being FinSequence of REAL n such that
A12: ( len g = m & ( for j being Nat st j in dom g holds
g . j = H2(j) ) ) from FINSEQ_2:sch_1();
A13: now__::_thesis:_for_j_being_Nat_st_j_in_dom_g_holds_
g_/._j_=_(z_/._j)_-_(z_/._(j_+_1))
let j be Nat; ::_thesis: ( j in dom g implies g /. j = (z /. j) - (z /. (j + 1)) )
assume A14: j in dom g ; ::_thesis: g /. j = (z /. j) - (z /. (j + 1))
then  g /. j = g . j by PARTFUN1:def_6;
hence  g /. j = (z /. j) - (z /. (j + 1)) by A14, A12; ::_thesis: verum
end;
A15: 1 <= m + 1 by NAT_1:11;
A16: (m + 1) -' 1 = (m + 1) - 1 by NAT_1:11, XREAL_1:233;
 1 in dom h by A6, A15;
then  h /. 1 = (y | ((m + 1) -' 1)) ^ (0* (1 -' 1)) by A7;
then  h /. 1 = (y | m) ^ (0* 0) by A16, XREAL_1:232;
then  h /. 1 = y ^ (0* 0) by A1, FINSEQ_1:58;
then A17: h /. 1 = y by FINSEQ_1:34;
A18: ( 1 <= len z & len z <= m + 1 ) by A9, NAT_1:14;
A19: (m + 1) -' (len z) = (m + 1) - (len z) by A9, XREAL_1:233;
A20: (len z) -' 1 = (len z) - 1 by A9, NAT_1:14, XREAL_1:233;
 len z in dom h by A6, A18;
then  h /. (len z) = (y | 0) ^ (0* ((len z) -' 1)) by A7, A19, A9;
then A21: h /. (len z) = 0* m by A20, A9, FINSEQ_1:34;
 1 <= m + 1 by NAT_1:11;
then  1 in Seg (m + 1) ;
then  1 in dom z by A9, FINSEQ_1:def_3;
then A22: z /. 1 = f /. (x + y) by A10, A17;
 len z in Seg (m + 1) by A9, FINSEQ_1:4;
then  len z in dom z by A9, FINSEQ_1:def_3;
then  z /. (len z) = f /. (x + (h /. (len z))) by A10;
then A23: z /. (len z) = f /. x by A21, EUCLID_4:1;
take  h ; ::_thesis: ex g being FinSequence of REAL n st
( len h = m + 1 & len g = m & ( for i being Nat st i in dom h holds
h /. i = (y | ((m + 1) -' i)) ^ (0* (i -' 1)) ) & ( for i being Nat st i in dom g holds
g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))) ) & ( for i being Nat
for hi being Element of REAL m st i in dom h & h /. i = hi holds
|.hi.| <= |.y.| ) & (f /. (x + y)) - (f /. x) = Sum g )
take  g ; ::_thesis: ( len h = m + 1 & len g = m & ( for i being Nat st i in dom h holds
h /. i = (y | ((m + 1) -' i)) ^ (0* (i -' 1)) ) & ( for i being Nat st i in dom g holds
g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))) ) & ( for i being Nat
for hi being Element of REAL m st i in dom h & h /. i = hi holds
|.hi.| <= |.y.| ) & (f /. (x + y)) - (f /. x) = Sum g )
thus  ( len h = m + 1 & len g = m ) by A6, A12, FINSEQ_1:def_3; ::_thesis: ( ( for i being Nat st i in dom h holds
h /. i = (y | ((m + 1) -' i)) ^ (0* (i -' 1)) ) & ( for i being Nat st i in dom g holds
g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))) ) & ( for i being Nat
for hi being Element of REAL m st i in dom h & h /. i = hi holds
|.hi.| <= |.y.| ) & (f /. (x + y)) - (f /. x) = Sum g )
A24: now__::_thesis:_for_i_being_Nat_st_i_in_dom_g_holds_
g_/._i_=_(f_/._(x_+_(h_/._i)))_-_(f_/._(x_+_(h_/._(i_+_1))))
let i be Nat; ::_thesis: ( i in dom g implies g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))) )
assume A25: i in dom g ; ::_thesis: g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1))))
then A26: i in Seg m by A12, FINSEQ_1:def_3;
 m <= m + 1 by NAT_1:11;
then A27: Seg m c= Seg (m + 1) by FINSEQ_1:5;
 dom h = dom z by A6, A9, FINSEQ_1:def_3;
then A28: z /. i = f /. (x + (h /. i)) by A10, A27, A6, A26;
 i in Seg m by A12, A25, FINSEQ_1:def_3;
then  ( 1 <= i & i <= m ) by FINSEQ_1:1;
then A29: i + 1 <= m + 1 by XREAL_1:6;
 1 <= i + 1 by NAT_1:11;
then  i + 1 in Seg (m + 1) by A29;
then  i + 1 in dom z by A9, FINSEQ_1:def_3;
then  z /. (i + 1) = f /. (x + (h /. (i + 1))) by A10;
hence  g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))) by A13, A25, A28; ::_thesis: verum
end;
 for i being Nat
for hi being Element of REAL m st i in dom h & h /. i = hi holds
|.hi.| <= |.y.|
proof
let i be Nat; ::_thesis: for hi being Element of REAL m st i in dom h & h /. i = hi holds
|.hi.| <= |.y.|
let hi be Element of REAL m; ::_thesis: ( i in dom h & h /. i = hi implies |.hi.| <= |.y.| )
assume  ( i in dom h & h /. i = hi ) ; ::_thesis: |.hi.| <= |.y.|
then A30: hi = (y | ((m + 1) -' i)) ^ (0* (i -' 1)) by A7;
A31: for j being Nat st j in Seg m holds
(sqr hi) . j <= (sqr y) . j
proof
let j be Nat; ::_thesis: ( j in Seg m implies (sqr hi) . j <= (sqr y) . j )
assume A32: j in Seg m ; ::_thesis: (sqr hi) . j <= (sqr y) . j
 len hi = m by CARD_1:def_7;
then A33: j in dom ((y | ((m + 1) -' i)) ^ (0* (i -' 1))) by A32, A30, FINSEQ_1:def_3;
percases ( j in dom (y | ((m + 1) -' i)) or ex k being Nat st
( k in dom (0* (i -' 1)) & j = (len (y | ((m + 1) -' i))) + k ) ) by A33, FINSEQ_1:25;
supposeA34: j in dom (y | ((m + 1) -' i)) ; ::_thesis: (sqr hi) . j <= (sqr y) . j
then  j in Seg (len (y | ((m + 1) -' i))) by FINSEQ_1:def_3;
then A35: j <= len (y | ((m + 1) -' i)) by FINSEQ_1:1;
A36: len (y | ((m + 1) -' i)) <= (m + 1) -' i by FINSEQ_5:17;
(sqr hi) . j = (sqrreal * hi) . j by RVSUM_1:def_8
.= sqrreal . (((y | ((m + 1) -' i)) ^ (0* (i -' 1))) . j) by A30, A33, FUNCT_1:13
.= sqrreal . ((y | ((m + 1) -' i)) . j) by A34, FINSEQ_1:def_7
.= sqrreal . (y . j) by A36, A35, FINSEQ_3:112, XXREAL_0:2
.= (y . j) ^2 by RVSUM_1:def_2
.= (sqr y) . j by VALUED_1:11 ;
hence  (sqr hi) . j <= (sqr y) . j ; ::_thesis: verum
end;
suppose ex k being Nat st
( k in dom (0* (i -' 1)) & j = (len (y | ((m + 1) -' i))) + k ) ; ::_thesis: (sqr hi) . j <= (sqr y) . j
then consider k being Nat such that
A37: ( k in dom (0* (i -' 1)) & j = (len (y | ((m + 1) -' i))) + k ) ;
A38: (sqr hi) . j = (sqrreal * hi) . j by RVSUM_1:def_8
.= sqrreal . (((y | ((m + 1) -' i)) ^ (0* (i -' 1))) . j) by A30, A33, FUNCT_1:13
.= sqrreal . ((0* (i -' 1)) . k) by A37, FINSEQ_1:def_7
.= ((0* (i -' 1)) . k) ^2 by RVSUM_1:def_2
.= 0 ;
(sqr y) . j = (y . j) ^2 by VALUED_1:11
.= (y . j) * (y . j) ;
hence  (sqr hi) . j <= (sqr y) . j by A38, XREAL_1:63; ::_thesis: verum
end;
end;
end;
 0 <= Sum (sqr hi) by RVSUM_1:86;
hence  |.hi.| <= |.y.| by A31, RVSUM_1:82, SQUARE_1:26; ::_thesis: verum
end;
hence  ( ( for i being Nat st i in dom h holds
h /. i = (y | ((m + 1) -' i)) ^ (0* (i -' 1)) ) & ( for i being Nat st i in dom g holds
g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))) ) & ( for i being Nat
for hi being Element of REAL m st i in dom h & h /. i = hi holds
|.hi.| <= |.y.| ) & (f /. (x + y)) - (f /. x) = Sum g ) by A7, A22, A23, A24, A9, A12, A13, Th26; ::_thesis: verum
end;

theorem Th29: :: PDIFF_7:29
for m being non empty Element of NAT
for f being PartFunc of (REAL m),(REAL 1) ex f0 being PartFunc of (REAL m),REAL st f = <>* f0
proof
let m be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL 1) ex f0 being PartFunc of (REAL m),REAL st f = <>* f0
let f be PartFunc of (REAL m),(REAL 1); ::_thesis: ex f0 being PartFunc of (REAL m),REAL st f = <>* f0
defpred S1[ set ] means $1 in dom f;
deffunc H1( set ) ->  Element of REAL = (proj (1,1)) . (f /. $1);
A1: for x being set st S1[x] holds
H1(x) in REAL ;
consider f0 being PartFunc of (REAL m),REAL such that
A2: ( ( for x being set holds
( x in dom f0 iff ( x in REAL m & S1[x] ) ) ) & ( for x being set st x in dom f0 holds
f0 . x = H1(x) ) ) from PARTFUN1:sch_3(A1);
take  f0 ; ::_thesis: f = <>* f0
 for x being set st x in dom f holds
x in dom f0 by A2;
then A3: dom f c= dom f0 by TARSKI:def_3;
 for x being set st x in dom f0 holds
x in dom f by A2;
then A4: dom f0 c= dom f by TARSKI:def_3;
then A5: dom f = dom f0 by A3, XBOOLE_0:def_10;
A6: rng f0 c= dom ((proj (1,1)) ") by PDIFF_1:2;
then A7: dom (((proj (1,1)) ") * f0) = dom f0 by RELAT_1:27;
 for x being Element of REAL m st x in dom (<>* f0) holds
(<>* f0) . x = f . x
proof
let x be Element of REAL m; ::_thesis: ( x in dom (<>* f0) implies (<>* f0) . x = f . x )
assume A8: x in dom (<>* f0) ; ::_thesis: (<>* f0) . x = f . x
then  (<>* f0) . x = ((proj (1,1)) ") . (f0 . x) by FUNCT_1:12;
then A9: (<>* f0) . x = ((proj (1,1)) ") . ((proj (1,1)) . (f /. x)) by A8, A7, A2;
 f /. x is Element of 1 -tuples_on REAL ;
then consider x0 being Element of REAL  such that
A10: f /. x = <*x0*> by FINSEQ_2:97;
 (proj (1,1)) . (f /. x) = x0 by A10, PDIFF_1:1;
then  (<>* f0) . x = f /. x by A9, A10, PDIFF_1:1;
hence  (<>* f0) . x = f . x by A7, A4, A8, PARTFUN1:def_6; ::_thesis: verum
end;
hence  f = <>* f0 by A6, A5, PARTFUN1:5, RELAT_1:27; ::_thesis: verum
end;

theorem Th30: :: PDIFF_7:30
for m, n being non empty Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for f0 being PartFunc of (REAL-NS m),(REAL-NS n)
for x being Element of REAL m
for x0 being Element of (REAL-NS m) st x in dom f & x = x0 & f = f0 holds
f /. x = f0 /. x0
proof
let m, n be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n)
for f0 being PartFunc of (REAL-NS m),(REAL-NS n)
for x being Element of REAL m
for x0 being Element of (REAL-NS m) st x in dom f & x = x0 & f = f0 holds
f /. x = f0 /. x0
let f be PartFunc of (REAL m),(REAL n); ::_thesis: for f0 being PartFunc of (REAL-NS m),(REAL-NS n)
for x being Element of REAL m
for x0 being Element of (REAL-NS m) st x in dom f & x = x0 & f = f0 holds
f /. x = f0 /. x0
let f0 be PartFunc of (REAL-NS m),(REAL-NS n); ::_thesis: for x being Element of REAL m
for x0 being Element of (REAL-NS m) st x in dom f & x = x0 & f = f0 holds
f /. x = f0 /. x0
let x be Element of REAL m; ::_thesis: for x0 being Element of (REAL-NS m) st x in dom f & x = x0 & f = f0 holds
f /. x = f0 /. x0
let x0 be Element of (REAL-NS m); ::_thesis: ( x in dom f & x = x0 & f = f0 implies f /. x = f0 /. x0 )
assume A1: ( x in dom f & x = x0 & f = f0 ) ; ::_thesis: f /. x = f0 /. x0
then  f /. x = f0 . x0 by PARTFUN1:def_6;
hence  f /. x = f0 /. x0 by A1, PARTFUN1:def_6; ::_thesis: verum
end;

definition
let m be non empty Element of NAT ;
let X be Subset of (REAL m);
attrX is open  means :Def3: :: PDIFF_7:def 3
 ex X0 being Subset of (REAL-NS m) st
( X0 = X & X0 is open );
end;

:: deftheorem Def3 defines open PDIFF_7:def_3_:_
 for m being non empty Element of NAT
for X being Subset of (REAL m) holds
( X is open iff ex X0 being Subset of (REAL-NS m) st
( X0 = X & X0 is open ) );

theorem Th31: :: PDIFF_7:31
for m being non empty Element of NAT
for X being Subset of (REAL m) holds
( X is open iff for x being Element of REAL m st x in X holds
ex r being Real st
( r > 0 &  {  y where y is Element of REAL m : |.(y - x).| < r  }  c= X ) )
proof
let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m) holds
( X is open iff for x being Element of REAL m st x in X holds
ex r being Real st
( r > 0 &  {  y where y is Element of REAL m : |.(y - x).| < r  }  c= X ) )
let X be Subset of (REAL m); ::_thesis: ( X is open iff for x being Element of REAL m st x in X holds
ex r being Real st
( r > 0 &  {  y where y is Element of REAL m : |.(y - x).| < r  }  c= X ) )
hereby  ::_thesis: ( ( for x being Element of REAL m st x in X holds
ex r being Real st
( r > 0 &  {  y where y is Element of REAL m : |.(y - x).| < r  }  c= X ) ) implies X is open )
assume  X is open ; ::_thesis: for x being Element of REAL m st x in X holds
ex d0 being Real st
( 0 < d0 &  {  y where y is Element of REAL m : |.(y - x).| < d0  }  c= X )
then consider VV0 being Subset of (REAL-NS m) such that
A1: ( X = VV0 & VV0 is open ) by Def3;
let x be Element of REAL m; ::_thesis: ( x in X implies ex d0 being Real st
( 0 < d0 &  {  y where y is Element of REAL m : |.(y - x).| < d0  }  c= X ) )
assume A2: x in X ; ::_thesis: ex d0 being Real st
( 0 < d0 &  {  y where y is Element of REAL m : |.(y - x).| < d0  }  c= X )
reconsider V0 = VV0 as Subset of (TopSpaceNorm (REAL-NS m)) ;
reconsider v0 = x as Point of (REAL-NS m) by REAL_NS1:def_4;
 V0 is open by A1, NORMSP_2:16;
then consider d0 being Real such that
A3: ( d0 > 0 &  {  w where w is Point of (REAL-NS m) : ||.(v0 - w).|| < d0  }  c= V0 ) by A2, A1, NORMSP_2:7;
take d0 = d0; ::_thesis: ( 0 < d0 &  {  y where y is Element of REAL m : |.(y - x).| < d0  }  c= X )
thus  0 < d0 by A3; ::_thesis:  {  y where y is Element of REAL m : |.(y - x).| < d0  }  c= X
thus   {  y where y is Element of REAL m : |.(y - x).| < d0  }  c= X ::_thesis: verum
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in  {  y where y is Element of REAL m : |.(y - x).| < d0  }  or z in X )
assume  z in  {  y where y is Element of REAL m : |.(y - x).| < d0  }  ; ::_thesis: z in X
then consider y being Element of REAL m such that
A4: ( z = y & |.(y - x).| < d0 ) ;
reconsider w = y as Point of (REAL-NS m) by REAL_NS1:def_4;
 |.(y - x).| = ||.(w - v0).|| by REAL_NS1:1, REAL_NS1:5;
then  ||.(v0 - w).|| < d0 by A4, NORMSP_1:7;
then  w in  {  w1 where w1 is Point of (REAL-NS m) : ||.(v0 - w1).|| < d0  }  ;
hence  z in X by A4, A1, A3; ::_thesis: verum
end;
end;
assume A5: for x being Element of REAL m st x in X holds
ex r being Real st
( r > 0 &  {  y where y is Element of REAL m : |.(y - x).| < r  }  c= X ) ; ::_thesis: X is open
reconsider VV0 = X as Subset of (REAL-NS m) by REAL_NS1:def_4;
reconsider V0 = VV0 as Subset of (TopSpaceNorm (REAL-NS m)) ;
 for v being Point of (REAL-NS m) st v in V0 holds
ex r being Real st
( r > 0 &  {  w where w is Point of (REAL-NS m) : ||.(v - w).|| < r  }  c= V0 )
proof
let v be Point of (REAL-NS m); ::_thesis: ( v in V0 implies ex r being Real st
( r > 0 &  {  w where w is Point of (REAL-NS m) : ||.(v - w).|| < r  }  c= V0 ) )
assume A6: v in V0 ; ::_thesis: ex r being Real st
( r > 0 &  {  w where w is Point of (REAL-NS m) : ||.(v - w).|| < r  }  c= V0 )
reconsider x = v as Element of REAL m by REAL_NS1:def_4;
consider d0 being Real such that
A7: ( d0 > 0 &  {  y where y is Element of REAL m : |.(y - x).| < d0  }  c= X ) by A5, A6;
take  d0 ; ::_thesis: ( d0 > 0 &  {  w where w is Point of (REAL-NS m) : ||.(v - w).|| < d0  }  c= V0 )
thus  0 < d0 by A7; ::_thesis:  {  w where w is Point of (REAL-NS m) : ||.(v - w).|| < d0  }  c= V0
thus   {  w where w is Point of (REAL-NS m) : ||.(v - w).|| < d0  }  c= V0 ::_thesis: verum
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in  {  w where w is Point of (REAL-NS m) : ||.(v - w).|| < d0  }  or z in V0 )
assume  z in  {  w where w is Point of (REAL-NS m) : ||.(v - w).|| < d0  }  ; ::_thesis: z in V0
then consider w being Point of (REAL-NS m) such that
A8: ( z = w & ||.(v - w).|| < d0 ) ;
reconsider y = w as Element of REAL m by REAL_NS1:def_4;
 |.(y - x).| = ||.(w - v).|| by REAL_NS1:1, REAL_NS1:5;
then  |.(y - x).| < d0 by A8, NORMSP_1:7;
then  y in  {  t where t is Element of REAL m : |.(t - x).| < d0  }  ;
hence  z in V0 by A8, A7; ::_thesis: verum
end;
end;
then  V0 is open by NORMSP_2:7;
then  VV0 is open by NORMSP_2:16;
hence  X is open by Def3; ::_thesis: verum
end;

definition
let m, n be non empty Element of NAT ;
let i be Element of NAT ;
let f be PartFunc of (REAL m),(REAL n);
let X be set ;
predf is_partial_differentiable_on X,i means :Def4: :: PDIFF_7:def 4
( X c= dom f & ( for x being Element of REAL m st x in X holds
f | X is_partial_differentiable_in x,i ) );
end;

:: deftheorem Def4 defines is_partial_differentiable_on PDIFF_7:def_4_:_
 for m, n being non empty Element of NAT
for i being Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for X being set holds
( f is_partial_differentiable_on X,i iff ( X c= dom f & ( for x being Element of REAL m st x in X holds
f | X is_partial_differentiable_in x,i ) ) );

theorem Th32: :: PDIFF_7:32
for i being Element of NAT
for X being set
for m, n being non empty Element of NAT
for f being PartFunc of (REAL m),(REAL n) st f is_partial_differentiable_on X,i holds
X is Subset of (REAL m)
proof
let i be Element of NAT ; ::_thesis: for X being set
for m, n being non empty Element of NAT
for f being PartFunc of (REAL m),(REAL n) st f is_partial_differentiable_on X,i holds
X is Subset of (REAL m)
let X be set ; ::_thesis: for m, n being non empty Element of NAT
for f being PartFunc of (REAL m),(REAL n) st f is_partial_differentiable_on X,i holds
X is Subset of (REAL m)
let m, n be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n) st f is_partial_differentiable_on X,i holds
X is Subset of (REAL m)
let f be PartFunc of (REAL m),(REAL n); ::_thesis: ( f is_partial_differentiable_on X,i implies X is Subset of (REAL m) )
assume  f is_partial_differentiable_on X,i ; ::_thesis: X is Subset of (REAL m)
then  X c= dom f by Def4;
hence  X is Subset of (REAL m) by XBOOLE_1:1; ::_thesis: verum
end;

theorem Th33: :: PDIFF_7:33
for m, n being non empty Element of NAT
for i being Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for g being PartFunc of (REAL-NS m),(REAL-NS n)
for Z being set st f = g holds
( f is_partial_differentiable_on Z,i iff g is_partial_differentiable_on Z,i )
proof
let m, n be non empty Element of NAT ; ::_thesis: for i being Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for g being PartFunc of (REAL-NS m),(REAL-NS n)
for Z being set st f = g holds
( f is_partial_differentiable_on Z,i iff g is_partial_differentiable_on Z,i )
let i be Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n)
for g being PartFunc of (REAL-NS m),(REAL-NS n)
for Z being set st f = g holds
( f is_partial_differentiable_on Z,i iff g is_partial_differentiable_on Z,i )
let f be PartFunc of (REAL m),(REAL n); ::_thesis: for g being PartFunc of (REAL-NS m),(REAL-NS n)
for Z being set st f = g holds
( f is_partial_differentiable_on Z,i iff g is_partial_differentiable_on Z,i )
let g be PartFunc of (REAL-NS m),(REAL-NS n); ::_thesis: for Z being set st f = g holds
( f is_partial_differentiable_on Z,i iff g is_partial_differentiable_on Z,i )
let Z be set ; ::_thesis: ( f = g implies ( f is_partial_differentiable_on Z,i iff g is_partial_differentiable_on Z,i ) )
assume A1: f = g ; ::_thesis: ( f is_partial_differentiable_on Z,i iff g is_partial_differentiable_on Z,i )
hereby  ::_thesis: ( g is_partial_differentiable_on Z,i implies f is_partial_differentiable_on Z,i )
assume A2: f is_partial_differentiable_on Z,i ; ::_thesis: g is_partial_differentiable_on Z,i
then A3: ( Z c= dom f & ( for x being Element of REAL m st x in Z holds
f | Z is_partial_differentiable_in x,i ) ) by Def4;
now__::_thesis:_for_y_being_Point_of_(REAL-NS_m)_st_y_in_Z_holds_
g_|_Z_is_partial_differentiable_in_y,i
let y be Point of (REAL-NS m); ::_thesis: ( y in Z implies g | Z is_partial_differentiable_in y,i )
assume A4: y in Z ; ::_thesis: g | Z is_partial_differentiable_in y,i
reconsider x = y as Element of REAL m by REAL_NS1:def_4;
 f | Z is_partial_differentiable_in x,i by A2, A4, Def4;
then  ex gZ being PartFunc of (REAL-NS m),(REAL-NS n) ex y being Point of (REAL-NS m) st
( f | Z = gZ & x = y & gZ is_partial_differentiable_in y,i ) by PDIFF_1:def_13;
hence  g | Z is_partial_differentiable_in y,i by A1; ::_thesis: verum
end;
hence  g is_partial_differentiable_on Z,i by A1, A3, PDIFF_1:def_19; ::_thesis: verum
end;
assume A5: g is_partial_differentiable_on Z,i ; ::_thesis: f is_partial_differentiable_on Z,i
then A6: ( Z c= dom g & ( for y being Point of (REAL-NS m) st y in Z holds
g | Z is_partial_differentiable_in y,i ) ) by PDIFF_1:def_19;
now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_Z_holds_
f_|_Z_is_partial_differentiable_in_x,i
let x be Element of REAL m; ::_thesis: ( x in Z implies f | Z is_partial_differentiable_in x,i )
assume A7: x in Z ; ::_thesis: f | Z is_partial_differentiable_in x,i
reconsider y = x as Point of (REAL-NS m) by REAL_NS1:def_4;
 g | Z is_partial_differentiable_in y,i by A5, A7, PDIFF_1:def_19;
hence  f | Z is_partial_differentiable_in x,i by A1, PDIFF_1:def_13; ::_thesis: verum
end;
hence  f is_partial_differentiable_on Z,i by A1, A6, Def4; ::_thesis: verum
end;

theorem Th34: :: PDIFF_7:34
for m, n being non empty Element of NAT
for i being Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for Z being Subset of (REAL m) st Z is open & 1 <= i & i <= m holds
( f is_partial_differentiable_on Z,i iff ( Z c= dom f & ( for x being Element of REAL m st x in Z holds
f is_partial_differentiable_in x,i ) ) )
proof
let m, n be non empty Element of NAT ; ::_thesis: for i being Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for Z being Subset of (REAL m) st Z is open & 1 <= i & i <= m holds
( f is_partial_differentiable_on Z,i iff ( Z c= dom f & ( for x being Element of REAL m st x in Z holds
f is_partial_differentiable_in x,i ) ) )
let i be Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n)
for Z being Subset of (REAL m) st Z is open & 1 <= i & i <= m holds
( f is_partial_differentiable_on Z,i iff ( Z c= dom f & ( for x being Element of REAL m st x in Z holds
f is_partial_differentiable_in x,i ) ) )
let f be PartFunc of (REAL m),(REAL n); ::_thesis: for Z being Subset of (REAL m) st Z is open & 1 <= i & i <= m holds
( f is_partial_differentiable_on Z,i iff ( Z c= dom f & ( for x being Element of REAL m st x in Z holds
f is_partial_differentiable_in x,i ) ) )
let Z be Subset of (REAL m); ::_thesis: ( Z is open & 1 <= i & i <= m implies ( f is_partial_differentiable_on Z,i iff ( Z c= dom f & ( for x being Element of REAL m st x in Z holds
f is_partial_differentiable_in x,i ) ) ) )
assume A1: ( Z is open & 1 <= i & i <= m ) ; ::_thesis: ( f is_partial_differentiable_on Z,i iff ( Z c= dom f & ( for x being Element of REAL m st x in Z holds
f is_partial_differentiable_in x,i ) ) )
then consider Z0 being Subset of (REAL-NS m) such that
A2: ( Z = Z0 & Z0 is open ) by Def3;
 ( the carrier of (REAL-NS m) = REAL m & the carrier of (REAL-NS n) = REAL n ) by REAL_NS1:def_4;
then reconsider g = f as PartFunc of (REAL-NS m),(REAL-NS n) ;
hereby  ::_thesis: ( Z c= dom f & ( for x being Element of REAL m st x in Z holds
f is_partial_differentiable_in x,i ) implies f is_partial_differentiable_on Z,i )
assume  f is_partial_differentiable_on Z,i ; ::_thesis: ( Z c= dom f & ( for x being Element of REAL m st x in Z holds
f is_partial_differentiable_in x,i ) )
then A3: g is_partial_differentiable_on Z0,i by A2, Th33;
hence  Z c= dom f by A1, A2, Th8; ::_thesis: for x being Element of REAL m st x in Z holds
f is_partial_differentiable_in x,i
thus  for x being Element of REAL m st x in Z holds
f is_partial_differentiable_in x,i ::_thesis: verum
proof
let x be Element of REAL m; ::_thesis: ( x in Z implies f is_partial_differentiable_in x,i )
assume A4: x in Z ; ::_thesis: f is_partial_differentiable_in x,i
reconsider y = x as Point of (REAL-NS m) by REAL_NS1:def_4;
 g is_partial_differentiable_in y,i by A2, A3, A4, A1, Th8;
hence  f is_partial_differentiable_in x,i by PDIFF_1:def_13; ::_thesis: verum
end;
end;
assume A5: ( Z c= dom f & ( for x being Element of REAL m st x in Z holds
f is_partial_differentiable_in x,i ) ) ; ::_thesis: f is_partial_differentiable_on Z,i
now__::_thesis:_for_y_being_Point_of_(REAL-NS_m)_st_y_in_Z0_holds_
g_is_partial_differentiable_in_y,i
let y be Point of (REAL-NS m); ::_thesis: ( y in Z0 implies g is_partial_differentiable_in y,i )
assume A6: y in Z0 ; ::_thesis: g is_partial_differentiable_in y,i
reconsider x = y as Element of REAL m by REAL_NS1:def_4;
 f is_partial_differentiable_in x,i by A2, A6, A5;
then  ex gZ being PartFunc of (REAL-NS m),(REAL-NS n) ex y being Point of (REAL-NS m) st
( f = gZ & x = y & gZ is_partial_differentiable_in y,i ) by PDIFF_1:def_13;
hence  g is_partial_differentiable_in y,i ; ::_thesis: verum
end;
then  g is_partial_differentiable_on Z0,i by A1, Th8, A2, A5;
hence  f is_partial_differentiable_on Z,i by Th33, A2; ::_thesis: verum
end;

definition
let m, n be non empty Element of NAT ;
let i be Element of NAT ;
let f be PartFunc of (REAL m),(REAL n);
let X be set ;
assume A1: f is_partial_differentiable_on X,i ;
funcf `partial| (X,i) ->  PartFunc of (REAL m),(REAL n) means :Def5: :: PDIFF_7:def 5
( dom it = X & ( for x being Element of REAL m st x in X holds
it /. x = partdiff (f,x,i) ) );
existence
 ex b1 being PartFunc of (REAL m),(REAL n) st
( dom b1 = X & ( for x being Element of REAL m st x in X holds
b1 /. x = partdiff (f,x,i) ) )
proof
deffunc H1( Element of REAL m) ->  Element of REAL n = partdiff (f,$1,i);
defpred S1[ Element of REAL m] means $1 in X;
consider F being PartFunc of (REAL m),(REAL n) such that
A2: ( ( for x being Element of REAL m holds
( x in dom F iff S1[x] ) ) & ( for x being Element of REAL m st x in dom F holds
F . x = H1(x) ) ) from SEQ_1:sch_3();
take  F ; ::_thesis: ( dom F = X & ( for x being Element of REAL m st x in X holds
F /. x = partdiff (f,x,i) ) )
now__::_thesis:_for_y_being_set_st_y_in_X_holds_
y_in_dom_F
A3: X is Subset of (REAL m) by A1, Th32;
let y be set ; ::_thesis: ( y in X implies y in dom F )
assume  y in X ; ::_thesis: y in dom F
hence  y in dom F by A2, A3; ::_thesis: verum
end;
then A4: X c= dom F by TARSKI:def_3;
 for y being set st y in dom F holds
y in X by A2;
then  dom F c= X by TARSKI:def_3;
hence  dom F = X by A4, XBOOLE_0:def_10; ::_thesis: for x being Element of REAL m st x in X holds
F /. x = partdiff (f,x,i)
hereby  ::_thesis: verum
let x be Element of REAL m; ::_thesis: ( x in X implies F /. x = partdiff (f,x,i) )
assume  x in X ; ::_thesis: F /. x = partdiff (f,x,i)
then A5: x in dom F by A2;
then  F . x = partdiff (f,x,i) by A2;
hence  F /. x = partdiff (f,x,i) by A5, PARTFUN1:def_6; ::_thesis: verum
end;
end;
uniqueness
 for b1, b2 being PartFunc of (REAL m),(REAL n) st dom b1 = X & ( for x being Element of REAL m st x in X holds
b1 /. x = partdiff (f,x,i) ) & dom b2 = X & ( for x being Element of REAL m st x in X holds
b2 /. x = partdiff (f,x,i) ) holds
b1 = b2
proof
let F, G be PartFunc of (REAL m),(REAL n); ::_thesis: ( dom F = X & ( for x being Element of REAL m st x in X holds
F /. x = partdiff (f,x,i) ) & dom G = X & ( for x being Element of REAL m st x in X holds
G /. x = partdiff (f,x,i) ) implies F = G )
assume that
A6: dom F = X and
A7: for x being Element of REAL m st x in X holds
F /. x = partdiff (f,x,i) and
A8: dom G = X and
A9: for x being Element of REAL m st x in X holds
G /. x = partdiff (f,x,i) ; ::_thesis: F = G
now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_dom_F_holds_
F_/._x_=_G_/._x
let x be Element of REAL m; ::_thesis: ( x in dom F implies F /. x = G /. x )
assume A10: x in dom F ; ::_thesis: F /. x = G /. x
then  F /. x = partdiff (f,x,i) by A6, A7;
hence  F /. x = G /. x by A6, A9, A10; ::_thesis: verum
end;
hence  F = G by A6, A8, PARTFUN2:1; ::_thesis: verum
end;
end;

:: deftheorem Def5 defines `partial| PDIFF_7:def_5_:_
 for m, n being non empty Element of NAT
for i being Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for X being set st f is_partial_differentiable_on X,i holds
for b6 being PartFunc of (REAL m),(REAL n) holds
( b6 = f `partial| (X,i) iff ( dom b6 = X & ( for x being Element of REAL m st x in X holds
b6 /. x = partdiff (f,x,i) ) ) );

definition
let m, n be non empty Element of NAT ;
let f be PartFunc of (REAL m),(REAL n);
let x0 be Element of REAL m;
predf is_continuous_in x0 means :Def6: :: PDIFF_7:def 6
 ex y0 being Point of (REAL-NS m) ex g being PartFunc of (REAL-NS m),(REAL-NS n) st
( x0 = y0 & f = g & g is_continuous_in y0 );
end;

:: deftheorem Def6 defines is_continuous_in PDIFF_7:def_6_:_
 for m, n being non empty Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for x0 being Element of REAL m holds
( f is_continuous_in x0 iff ex y0 being Point of (REAL-NS m) ex g being PartFunc of (REAL-NS m),(REAL-NS n) st
( x0 = y0 & f = g & g is_continuous_in y0 ) );

theorem Th35: :: PDIFF_7:35
for m, n being non empty Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for g being PartFunc of (REAL-NS m),(REAL-NS n)
for x being Element of REAL m
for y being Point of (REAL-NS m) st f = g & x = y holds
( f is_continuous_in x iff g is_continuous_in y )
proof
let m, n be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n)
for g being PartFunc of (REAL-NS m),(REAL-NS n)
for x being Element of REAL m
for y being Point of (REAL-NS m) st f = g & x = y holds
( f is_continuous_in x iff g is_continuous_in y )
let f be PartFunc of (REAL m),(REAL n); ::_thesis: for g being PartFunc of (REAL-NS m),(REAL-NS n)
for x being Element of REAL m
for y being Point of (REAL-NS m) st f = g & x = y holds
( f is_continuous_in x iff g is_continuous_in y )
let g be PartFunc of (REAL-NS m),(REAL-NS n); ::_thesis: for x being Element of REAL m
for y being Point of (REAL-NS m) st f = g & x = y holds
( f is_continuous_in x iff g is_continuous_in y )
let x be Element of REAL m; ::_thesis: for y being Point of (REAL-NS m) st f = g & x = y holds
( f is_continuous_in x iff g is_continuous_in y )
let y be Point of (REAL-NS m); ::_thesis: ( f = g & x = y implies ( f is_continuous_in x iff g is_continuous_in y ) )
assume A1: ( f = g & x = y ) ; ::_thesis: ( f is_continuous_in x iff g is_continuous_in y )
hereby  ::_thesis: ( g is_continuous_in y implies f is_continuous_in x )
assume  f is_continuous_in x ; ::_thesis: g is_continuous_in y
then  ex y0 being Point of (REAL-NS m) ex g being PartFunc of (REAL-NS m),(REAL-NS n) st
( x = y0 & f = g & g is_continuous_in y0 ) by Def6;
hence  g is_continuous_in y by A1; ::_thesis: verum
end;
assume  g is_continuous_in y ; ::_thesis: f is_continuous_in x
hence  f is_continuous_in x by Def6, A1; ::_thesis: verum
end;

theorem :: PDIFF_7:36
for m, n being non empty Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for x0 being Element of REAL m holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in dom f & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) )
proof
let m, n be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n)
for x0 being Element of REAL m holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in dom f & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) )
let f be PartFunc of (REAL m),(REAL n); ::_thesis: for x0 being Element of REAL m holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in dom f & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) )
let x0 be Element of REAL m; ::_thesis: ( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in dom f & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) )
 ( the carrier of (REAL-NS m) = REAL m & the carrier of (REAL-NS n) = REAL n ) by REAL_NS1:def_4;
then reconsider g = f as PartFunc of (REAL-NS m),(REAL-NS n) ;
reconsider y0 = x0 as Point of (REAL-NS m) by REAL_NS1:def_4;
hereby  ::_thesis: ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in dom f & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) implies f is_continuous_in x0 )
assume  f is_continuous_in x0 ; ::_thesis: ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in dom f & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) )
then A1: g is_continuous_in y0 by Th35;
then A2: ( y0 in dom g & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in dom g & ||.(y1 - y0).|| < s holds
||.((g /. y1) - (g /. y0)).|| < r ) ) ) ) by NFCONT_1:7;
thus  x0 in dom f by A1, NFCONT_1:7; ::_thesis: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in dom f & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
thus  for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in dom f & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ::_thesis: verum
proof
let r be Real; ::_thesis: ( 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in dom f & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) )
assume  0 < r ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in dom f & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
then consider s being Real such that
A3: ( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in dom g & ||.(y1 - y0).|| < s holds
||.((g /. y1) - (g /. y0)).|| < r ) ) by A1, NFCONT_1:7;
take  s ; ::_thesis: ( 0 < s & ( for x1 being Element of REAL m st x1 in dom f & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
thus  0 < s by A3; ::_thesis: for x1 being Element of REAL m st x1 in dom f & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r
hereby  ::_thesis: verum
let x1 be Element of REAL m; ::_thesis: ( x1 in dom f & |.(x1 - x0).| < s implies |.((f /. x1) - (f /. x0)).| < r )
assume A4: ( x1 in dom f & |.(x1 - x0).| < s ) ; ::_thesis: |.((f /. x1) - (f /. x0)).| < r
reconsider y1 = x1 as Point of (REAL-NS m) by REAL_NS1:def_4;
 ( y1 in dom g & ||.(y1 - y0).|| < s ) by A4, REAL_NS1:1, REAL_NS1:5;
then A5: ||.((g /. y1) - (g /. y0)).|| < r by A3;
 ( g /. y1 = f /. x1 & g /. y0 = f /. x0 ) by A2, Th30, A4;
hence  |.((f /. x1) - (f /. x0)).| < r by A5, REAL_NS1:1, REAL_NS1:5; ::_thesis: verum
end;
end;
end;
assume A6: ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in dom f & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) ; ::_thesis: f is_continuous_in x0
reconsider y0 = x0 as Point of (REAL-NS m) by REAL_NS1:def_4;
 for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in dom g & ||.(y1 - y0).|| < s holds
||.((g /. y1) - (g /. y0)).|| < r ) )
proof
let r be Real; ::_thesis: ( 0 < r implies ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in dom g & ||.(y1 - y0).|| < s holds
||.((g /. y1) - (g /. y0)).|| < r ) ) )
assume  0 < r ; ::_thesis: ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in dom g & ||.(y1 - y0).|| < s holds
||.((g /. y1) - (g /. y0)).|| < r ) )
then consider s being Real such that
A7: ( 0 < s & ( for x1 being Element of REAL m st x1 in dom f & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) by A6;
take  s ; ::_thesis: ( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in dom g & ||.(y1 - y0).|| < s holds
||.((g /. y1) - (g /. y0)).|| < r ) )
thus  0 < s by A7; ::_thesis: for y1 being Point of (REAL-NS m) st y1 in dom g & ||.(y1 - y0).|| < s holds
||.((g /. y1) - (g /. y0)).|| < r
hereby  ::_thesis: verum
let y1 be Point of (REAL-NS m); ::_thesis: ( y1 in dom g & ||.(y1 - y0).|| < s implies ||.((g /. y1) - (g /. y0)).|| < r )
assume A8: ( y1 in dom g & ||.(y1 - y0).|| < s ) ; ::_thesis: ||.((g /. y1) - (g /. y0)).|| < r
reconsider x1 = y1 as Element of REAL m by REAL_NS1:def_4;
 ( x1 in dom f & |.(x1 - x0).| < s ) by A8, REAL_NS1:1, REAL_NS1:5;
then A9: |.((f /. x1) - (f /. x0)).| < r by A7;
 ( g /. y1 = f /. x1 & g /. y0 = f /. x0 ) by A8, A6, Th30;
hence  ||.((g /. y1) - (g /. y0)).|| < r by A9, REAL_NS1:1, REAL_NS1:5; ::_thesis: verum
end;
end;
then  g is_continuous_in y0 by A6, NFCONT_1:7;
hence  f is_continuous_in x0 by Th35; ::_thesis: verum
end;

definition
let m, n be non empty Element of NAT ;
let f be PartFunc of (REAL m),(REAL n);
let X be set ;
predf is_continuous_on X means :Def7: :: PDIFF_7:def 7
( X c= dom f & ( for x0 being Element of REAL m st x0 in X holds
f | X is_continuous_in x0 ) );
end;

:: deftheorem Def7 defines is_continuous_on PDIFF_7:def_7_:_
 for m, n being non empty Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for X being set holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Element of REAL m st x0 in X holds
f | X is_continuous_in x0 ) ) );

theorem Th37: :: PDIFF_7:37
for m, n being non empty Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for g being PartFunc of (REAL-NS m),(REAL-NS n)
for X being set st f = g holds
( f is_continuous_on X iff g is_continuous_on X )
proof
let m, n be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n)
for g being PartFunc of (REAL-NS m),(REAL-NS n)
for X being set st f = g holds
( f is_continuous_on X iff g is_continuous_on X )
let f be PartFunc of (REAL m),(REAL n); ::_thesis: for g being PartFunc of (REAL-NS m),(REAL-NS n)
for X being set st f = g holds
( f is_continuous_on X iff g is_continuous_on X )
let g be PartFunc of (REAL-NS m),(REAL-NS n); ::_thesis: for X being set st f = g holds
( f is_continuous_on X iff g is_continuous_on X )
let X be set ; ::_thesis: ( f = g implies ( f is_continuous_on X iff g is_continuous_on X ) )
assume A1: f = g ; ::_thesis: ( f is_continuous_on X iff g is_continuous_on X )
hereby  ::_thesis: ( g is_continuous_on X implies f is_continuous_on X )
assume A2: f is_continuous_on X ; ::_thesis: g is_continuous_on X
then A3: ( X c= dom f & ( for x0 being Element of REAL m st x0 in X holds
f | X is_continuous_in x0 ) ) by Def7;
now__::_thesis:_for_y0_being_Point_of_(REAL-NS_m)_st_y0_in_X_holds_
g_|_X_is_continuous_in_y0
let y0 be Point of (REAL-NS m); ::_thesis: ( y0 in X implies g | X is_continuous_in y0 )
assume A4: y0 in X ; ::_thesis: g | X is_continuous_in y0
reconsider x0 = y0 as Element of REAL m by REAL_NS1:def_4;
 f | X is_continuous_in x0 by A2, A4, Def7;
hence  g | X is_continuous_in y0 by A1, Th35; ::_thesis: verum
end;
hence  g is_continuous_on X by A3, A1, NFCONT_1:def_7; ::_thesis: verum
end;
assume A5: g is_continuous_on X ; ::_thesis: f is_continuous_on X
then A6: ( X c= dom g & ( for y0 being Point of (REAL-NS m) st y0 in X holds
g | X is_continuous_in y0 ) ) by NFCONT_1:def_7;
now__::_thesis:_for_x0_being_Element_of_REAL_m_st_x0_in_X_holds_
f_|_X_is_continuous_in_x0
let x0 be Element of REAL m; ::_thesis: ( x0 in X implies f | X is_continuous_in x0 )
assume A7: x0 in X ; ::_thesis: f | X is_continuous_in x0
reconsider y0 = x0 as Point of (REAL-NS m) by REAL_NS1:def_4;
 g | X is_continuous_in y0 by A5, A7, NFCONT_1:def_7;
hence  f | X is_continuous_in x0 by A1, Th35; ::_thesis: verum
end;
hence  f is_continuous_on X by A6, A1, Def7; ::_thesis: verum
end;

theorem Th38: :: PDIFF_7:38
for m, n being non empty Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for X being set holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) )
proof
let m, n be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n)
for X being set holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) )
let f be PartFunc of (REAL m),(REAL n); ::_thesis: for X being set holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) )
let X be set ; ::_thesis: ( f is_continuous_on X iff ( X c= dom f & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) )
 ( the carrier of (REAL-NS m) = REAL m & the carrier of (REAL-NS n) = REAL n ) by REAL_NS1:def_4;
then reconsider g = f as PartFunc of (REAL-NS m),(REAL-NS n) ;
hereby  ::_thesis: ( X c= dom f & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) implies f is_continuous_on X )
assume  f is_continuous_on X ; ::_thesis: ( X c= dom f & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) )
then A1: g is_continuous_on X by Th37;
hence A2: X c= dom f by NFCONT_1:19; ::_thesis: for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
thus  for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ::_thesis: verum
proof
let x0 be Element of REAL m; ::_thesis: for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
let r be Real; ::_thesis: ( x0 in X & 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) )
reconsider y0 = x0 as Point of (REAL-NS m) by REAL_NS1:def_4;
assume A3: ( x0 in X & 0 < r ) ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
then consider s being Real such that
A4: ( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < s holds
||.((g /. y1) - (g /. y0)).|| < r ) ) by A1, NFCONT_1:19;
take  s ; ::_thesis: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
thus  0 < s by A4; ::_thesis: for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r
hereby  ::_thesis: verum
let x1 be Element of REAL m; ::_thesis: ( x1 in X & |.(x1 - x0).| < s implies |.((f /. x1) - (f /. x0)).| < r )
assume A5: ( x1 in X & |.(x1 - x0).| < s ) ; ::_thesis: |.((f /. x1) - (f /. x0)).| < r
reconsider y1 = x1 as Point of (REAL-NS m) by REAL_NS1:def_4;
 ( y1 in X & ||.(y1 - y0).|| < s ) by A5, REAL_NS1:1, REAL_NS1:5;
then A6: ||.((g /. y1) - (g /. y0)).|| < r by A4;
 ( g /. y1 = f /. x1 & g /. y0 = f /. x0 ) by A5, A2, A3, Th30;
hence  |.((f /. x1) - (f /. x0)).| < r by A6, REAL_NS1:1, REAL_NS1:5; ::_thesis: verum
end;
end;
end;
assume A7: ( X c= dom f & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) ; ::_thesis: f is_continuous_on X
 for y0 being Point of (REAL-NS m)
for r being Real st y0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < s holds
||.((g /. y1) - (g /. y0)).|| < r ) )
proof
let y0 be Point of (REAL-NS m); ::_thesis: for r being Real st y0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < s holds
||.((g /. y1) - (g /. y0)).|| < r ) )
let r be Real; ::_thesis: ( y0 in X & 0 < r implies ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < s holds
||.((g /. y1) - (g /. y0)).|| < r ) ) )
reconsider x0 = y0 as Element of REAL m by REAL_NS1:def_4;
assume A8: ( y0 in X & 0 < r ) ; ::_thesis: ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < s holds
||.((g /. y1) - (g /. y0)).|| < r ) )
then consider s being Real such that
A9: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) by A7;
take  s ; ::_thesis: ( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < s holds
||.((g /. y1) - (g /. y0)).|| < r ) )
thus  0 < s by A9; ::_thesis: for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < s holds
||.((g /. y1) - (g /. y0)).|| < r
hereby  ::_thesis: verum
let y1 be Point of (REAL-NS m); ::_thesis: ( y1 in X & ||.(y1 - y0).|| < s implies ||.((g /. y1) - (g /. y0)).|| < r )
assume A10: ( y1 in X & ||.(y1 - y0).|| < s ) ; ::_thesis: ||.((g /. y1) - (g /. y0)).|| < r
reconsider x1 = y1 as Element of REAL m by REAL_NS1:def_4;
 ( x1 in X & |.(x1 - x0).| < s ) by A10, REAL_NS1:1, REAL_NS1:5;
then A11: |.((f /. x1) - (f /. x0)).| < r by A9;
 ( g /. y1 = f /. x1 & g /. y0 = f /. x0 ) by A10, A7, A8, Th30;
hence  ||.((g /. y1) - (g /. y0)).|| < r by A11, REAL_NS1:1, REAL_NS1:5; ::_thesis: verum
end;
end;
then  g is_continuous_on X by A7, NFCONT_1:19;
hence  f is_continuous_on X by Th37; ::_thesis: verum
end;

theorem Th39: :: PDIFF_7:39
for m being non empty Element of NAT
for x, y being Element of REAL m
for i being Element of NAT
for xi being Real st 1 <= i & i <= m & y = (reproj (i,x)) . xi holds
(proj (i,m)) . y = xi
proof
let m be non empty Element of NAT ; ::_thesis: for x, y being Element of REAL m
for i being Element of NAT
for xi being Real st 1 <= i & i <= m & y = (reproj (i,x)) . xi holds
(proj (i,m)) . y = xi
let x, y be Element of REAL m; ::_thesis: for i being Element of NAT
for xi being Real st 1 <= i & i <= m & y = (reproj (i,x)) . xi holds
(proj (i,m)) . y = xi
let i be Element of NAT ; ::_thesis: for xi being Real st 1 <= i & i <= m & y = (reproj (i,x)) . xi holds
(proj (i,m)) . y = xi
let xi be Real; ::_thesis: ( 1 <= i & i <= m & y = (reproj (i,x)) . xi implies (proj (i,m)) . y = xi )
assume A1: ( 1 <= i & i <= m & y = (reproj (i,x)) . xi ) ; ::_thesis: (proj (i,m)) . y = xi
then A2: y = Replace (x,i,xi) by PDIFF_1:def_5;
A3: ( len x = m & len y = m ) by CARD_1:def_7;
then A4: i in dom y by A1, FINSEQ_3:25;
 y /. i = xi by A1, A2, A3, FINSEQ_7:8;
then  y . i = xi by A4, PARTFUN1:def_6;
hence  (proj (i,m)) . y = xi by PDIFF_1:def_1; ::_thesis: verum
end;

theorem Th40: :: PDIFF_7:40
for m being non empty Element of NAT
for f being PartFunc of (REAL m),REAL
for x, y being Element of REAL m
for i being Element of NAT
for xi being Real st 1 <= i & i <= m & y = (reproj (i,x)) . xi holds
reproj (i,x) = reproj (i,y)
proof
let m be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),REAL
for x, y being Element of REAL m
for i being Element of NAT
for xi being Real st 1 <= i & i <= m & y = (reproj (i,x)) . xi holds
reproj (i,x) = reproj (i,y)
let f be PartFunc of (REAL m),REAL; ::_thesis: for x, y being Element of REAL m
for i being Element of NAT
for xi being Real st 1 <= i & i <= m & y = (reproj (i,x)) . xi holds
reproj (i,x) = reproj (i,y)
let x, y be Element of REAL m; ::_thesis: for i being Element of NAT
for xi being Real st 1 <= i & i <= m & y = (reproj (i,x)) . xi holds
reproj (i,x) = reproj (i,y)
let i be Element of NAT ; ::_thesis: for xi being Real st 1 <= i & i <= m & y = (reproj (i,x)) . xi holds
reproj (i,x) = reproj (i,y)
let xi be Real; ::_thesis: ( 1 <= i & i <= m & y = (reproj (i,x)) . xi implies reproj (i,x) = reproj (i,y) )
assume A1: ( 1 <= i & i <= m & y = (reproj (i,x)) . xi ) ; ::_thesis: reproj (i,x) = reproj (i,y)
then A2: y = Replace (x,i,xi) by PDIFF_1:def_5;
A3: ( len x = m & len y = m ) by CARD_1:def_7;
then A4: y = ((x | (i -' 1)) ^ <*xi*>) ^ (x /^ i) by A1, A2, FINSEQ_7:def_1;
A5: dom (reproj (i,x)) = REAL by PDIFF_1:def_5
.= dom (reproj (i,y)) by PDIFF_1:def_5 ;
 for r being set st r in dom (reproj (i,x)) holds
(reproj (i,x)) . r = (reproj (i,y)) . r
proof
let r be set ; ::_thesis: ( r in dom (reproj (i,x)) implies (reproj (i,x)) . r = (reproj (i,y)) . r )
assume A6: r in dom (reproj (i,x)) ; ::_thesis: (reproj (i,x)) . r = (reproj (i,y)) . r
A7: ( i -' 1 <= len y & i -' 1 <= len x ) by A1, A3, NAT_D:44;
reconsider r1 = r as Real by A6;
 (reproj (i,x)) . r = Replace (x,i,r1) by PDIFF_1:def_5;
then A8: (reproj (i,x)) . r = ((x | (i -' 1)) ^ <*r1*>) ^ (x /^ i) by A1, A3, FINSEQ_7:def_1;
 (reproj (i,y)) . r = Replace (y,i,r1) by PDIFF_1:def_5;
then A9: (reproj (i,y)) . r = ((y | (i -' 1)) ^ <*r1*>) ^ (y /^ i) by A1, A3, FINSEQ_7:def_1;
A10: dom (y | (i -' 1)) = Seg (i -' 1) by A7, FINSEQ_1:17;
then A11: dom (y | (i -' 1)) = dom (x | (i -' 1)) by A7, FINSEQ_1:17;
A12: for n being Nat st n in dom (y | (i -' 1)) holds
(y | (i -' 1)) /. n = (x | (i -' 1)) /. n
proof
let n be Nat; ::_thesis: ( n in dom (y | (i -' 1)) implies (y | (i -' 1)) /. n = (x | (i -' 1)) /. n )
assume A13: n in dom (y | (i -' 1)) ; ::_thesis: (y | (i -' 1)) /. n = (x | (i -' 1)) /. n
then  n in Seg (len (x | (i -' 1))) by A7, A10, FINSEQ_1:17;
then A14: n <= len (x | (i -' 1)) by FINSEQ_1:1;
A15: len (x | (i -' 1)) <= i -' 1 by FINSEQ_5:17;
A16: ( 1 <= n & n <= len (x | (i -' 1)) ) by A13, A11, FINSEQ_3:25;
(y | (i -' 1)) /. n = (y | (i -' 1)) . n by A13, PARTFUN1:def_6
.= (((x | (i -' 1)) ^ <*xi*>) ^ (x /^ i)) . n by A4, A15, A14, FINSEQ_3:112, XXREAL_0:2
.= ((x | (i -' 1)) ^ (<*xi*> ^ (x /^ i))) . n by FINSEQ_1:32
.= (x | (i -' 1)) . n by A16, FINSEQ_1:64
.= (x | (i -' 1)) /. n by A13, A11, PARTFUN1:def_6 ;
hence  (y | (i -' 1)) /. n = (x | (i -' 1)) /. n ; ::_thesis: verum
end;
A17: ( len (y /^ i) = (len y) -' i & len (x /^ i) = (len x) -' i ) by RFINSEQ:29;
 for n being Nat st 1 <= n & n <= len (y /^ i) holds
(y /^ i) . n = (x /^ i) . n
proof
let n be Nat; ::_thesis: ( 1 <= n & n <= len (y /^ i) implies (y /^ i) . n = (x /^ i) . n )
assume A18: ( 1 <= n & n <= len (y /^ i) ) ; ::_thesis: (y /^ i) . n = (x /^ i) . n
then A19: ( n in dom (y /^ i) & n in dom (x /^ i) ) by A17, A3, FINSEQ_3:25;
A20: len (x | (i -' 1)) = i -' 1 by A1, A3, FINSEQ_1:59, NAT_D:44;
A21: len <*xi*> = 1 by FINSEQ_1:39;
 i - 1 >= 0 by A1, XREAL_1:48;
then  i -' 1 = i - 1 by XREAL_0:def_2;
then A22: len ((x | (i -' 1)) ^ <*xi*>) = (i - 1) + 1 by A20, A21, FINSEQ_1:22
.= i ;
(y /^ i) . n = y . (i + n) by A19, FINSEQ_7:4
.= (x /^ i) . n by A18, A17, A3, A4, A22, FINSEQ_1:65 ;
hence  (y /^ i) . n = (x /^ i) . n ; ::_thesis: verum
end;
then  y /^ i = x /^ i by A17, A3, FINSEQ_1:14;
hence  (reproj (i,x)) . r = (reproj (i,y)) . r by A8, A9, A11, A12, FINSEQ_5:12; ::_thesis: verum
end;
hence  reproj (i,x) = reproj (i,y) by A5, FUNCT_1:2; ::_thesis: verum
end;

theorem Th41: :: PDIFF_7:41
for m being non empty Element of NAT
for f being PartFunc of (REAL m),REAL
for g being PartFunc of REAL,REAL
for x, y being Element of REAL m
for i being Element of NAT
for xi being Real st 1 <= i & i <= m & y = (reproj (i,x)) . xi & g = f * (reproj (i,x)) holds
diff (g,xi) = partdiff (f,y,i)
proof
let m be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),REAL
for g being PartFunc of REAL,REAL
for x, y being Element of REAL m
for i being Element of NAT
for xi being Real st 1 <= i & i <= m & y = (reproj (i,x)) . xi & g = f * (reproj (i,x)) holds
diff (g,xi) = partdiff (f,y,i)
let f be PartFunc of (REAL m),REAL; ::_thesis: for g being PartFunc of REAL,REAL
for x, y being Element of REAL m
for i being Element of NAT
for xi being Real st 1 <= i & i <= m & y = (reproj (i,x)) . xi & g = f * (reproj (i,x)) holds
diff (g,xi) = partdiff (f,y,i)
let g be PartFunc of REAL,REAL; ::_thesis: for x, y being Element of REAL m
for i being Element of NAT
for xi being Real st 1 <= i & i <= m & y = (reproj (i,x)) . xi & g = f * (reproj (i,x)) holds
diff (g,xi) = partdiff (f,y,i)
let x, y be Element of REAL m; ::_thesis: for i being Element of NAT
for xi being Real st 1 <= i & i <= m & y = (reproj (i,x)) . xi & g = f * (reproj (i,x)) holds
diff (g,xi) = partdiff (f,y,i)
let i be Element of NAT ; ::_thesis: for xi being Real st 1 <= i & i <= m & y = (reproj (i,x)) . xi & g = f * (reproj (i,x)) holds
diff (g,xi) = partdiff (f,y,i)
let xi be Real; ::_thesis: ( 1 <= i & i <= m & y = (reproj (i,x)) . xi & g = f * (reproj (i,x)) implies diff (g,xi) = partdiff (f,y,i) )
assume A1: ( 1 <= i & i <= m & y = (reproj (i,x)) . xi & g = f * (reproj (i,x)) ) ; ::_thesis: diff (g,xi) = partdiff (f,y,i)
then  ( reproj (i,x) = reproj (i,y) & (proj (i,m)) . y = xi ) by Th39, Th40;
hence  partdiff (f,y,i) = diff (g,xi) by A1; ::_thesis: verum
end;

theorem Th42: :: PDIFF_7:42
for m being non empty Element of NAT
for f being PartFunc of (REAL m),REAL
for p, q being Real
for x being Element of REAL m
for i being Element of NAT st 1 <= i & i <= m & p < q & ( for h being Real st h in [.p,q.] holds
(reproj (i,x)) . h in dom f ) & ( for h being Real st h in [.p,q.] holds
f is_partial_differentiable_in (reproj (i,x)) . h,i ) holds
ex r being Real ex y being Element of REAL m st
( r in ].p,q.[ & y = (reproj (i,x)) . r & (f /. ((reproj (i,x)) . q)) - (f /. ((reproj (i,x)) . p)) = (q - p) * (partdiff (f,y,i)) )
proof
let m be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),REAL
for p, q being Real
for x being Element of REAL m
for i being Element of NAT st 1 <= i & i <= m & p < q & ( for h being Real st h in [.p,q.] holds
(reproj (i,x)) . h in dom f ) & ( for h being Real st h in [.p,q.] holds
f is_partial_differentiable_in (reproj (i,x)) . h,i ) holds
ex r being Real ex y being Element of REAL m st
( r in ].p,q.[ & y = (reproj (i,x)) . r & (f /. ((reproj (i,x)) . q)) - (f /. ((reproj (i,x)) . p)) = (q - p) * (partdiff (f,y,i)) )
let f be PartFunc of (REAL m),REAL; ::_thesis: for p, q being Real
for x being Element of REAL m
for i being Element of NAT st 1 <= i & i <= m & p < q & ( for h being Real st h in [.p,q.] holds
(reproj (i,x)) . h in dom f ) & ( for h being Real st h in [.p,q.] holds
f is_partial_differentiable_in (reproj (i,x)) . h,i ) holds
ex r being Real ex y being Element of REAL m st
( r in ].p,q.[ & y = (reproj (i,x)) . r & (f /. ((reproj (i,x)) . q)) - (f /. ((reproj (i,x)) . p)) = (q - p) * (partdiff (f,y,i)) )
let p, q be Real; ::_thesis: for x being Element of REAL m
for i being Element of NAT st 1 <= i & i <= m & p < q & ( for h being Real st h in [.p,q.] holds
(reproj (i,x)) . h in dom f ) & ( for h being Real st h in [.p,q.] holds
f is_partial_differentiable_in (reproj (i,x)) . h,i ) holds
ex r being Real ex y being Element of REAL m st
( r in ].p,q.[ & y = (reproj (i,x)) . r & (f /. ((reproj (i,x)) . q)) - (f /. ((reproj (i,x)) . p)) = (q - p) * (partdiff (f,y,i)) )
let x be Element of REAL m; ::_thesis: for i being Element of NAT st 1 <= i & i <= m & p < q & ( for h being Real st h in [.p,q.] holds
(reproj (i,x)) . h in dom f ) & ( for h being Real st h in [.p,q.] holds
f is_partial_differentiable_in (reproj (i,x)) . h,i ) holds
ex r being Real ex y being Element of REAL m st
( r in ].p,q.[ & y = (reproj (i,x)) . r & (f /. ((reproj (i,x)) . q)) - (f /. ((reproj (i,x)) . p)) = (q - p) * (partdiff (f,y,i)) )
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m & p < q & ( for h being Real st h in [.p,q.] holds
(reproj (i,x)) . h in dom f ) & ( for h being Real st h in [.p,q.] holds
f is_partial_differentiable_in (reproj (i,x)) . h,i ) implies ex r being Real ex y being Element of REAL m st
( r in ].p,q.[ & y = (reproj (i,x)) . r & (f /. ((reproj (i,x)) . q)) - (f /. ((reproj (i,x)) . p)) = (q - p) * (partdiff (f,y,i)) ) )
assume A1: ( 1 <= i & i <= m & p < q & ( for h being Real st h in [.p,q.] holds
(reproj (i,x)) . h in dom f ) & ( for h being Real st h in [.p,q.] holds
f is_partial_differentiable_in (reproj (i,x)) . h,i ) ) ; ::_thesis: ex r being Real ex y being Element of REAL m st
( r in ].p,q.[ & y = (reproj (i,x)) . r & (f /. ((reproj (i,x)) . q)) - (f /. ((reproj (i,x)) . p)) = (q - p) * (partdiff (f,y,i)) )
set g = f * (reproj (i,x));
now__::_thesis:_for_h_being_set_st_h_in_[.p,q.]_holds_
h_in_dom_(f_*_(reproj_(i,x)))
let h be set ; ::_thesis: ( h in [.p,q.] implies h in dom (f * (reproj (i,x))) )
assume A2: h in [.p,q.] ; ::_thesis: h in dom (f * (reproj (i,x)))
then reconsider h1 = h as Element of REAL ;
A3: dom (reproj (i,x)) = REAL by PDIFF_1:def_5;
 (reproj (i,x)) . h1 in dom f by A1, A2;
hence  h in dom (f * (reproj (i,x))) by A3, FUNCT_1:11; ::_thesis: verum
end;
then A4: [.p,q.] c= dom (f * (reproj (i,x))) by TARSKI:def_3;
A5: now__::_thesis:_for_x0_being_Real_st_x0_in_[.p,q.]_holds_
f_*_(reproj_(i,x))_is_differentiable_in_x0
let x0 be Real; ::_thesis: ( x0 in [.p,q.] implies f * (reproj (i,x)) is_differentiable_in x0 )
assume A6: x0 in [.p,q.] ; ::_thesis: f * (reproj (i,x)) is_differentiable_in x0
set y = (reproj (i,x)) . x0;
A7: (proj (i,m)) . ((reproj (i,x)) . x0) = x0 by Th39, A1;
 f is_partial_differentiable_in (reproj (i,x)) . x0,i by A1, A6;
then  f * (reproj (i,((reproj (i,x)) . x0))) is_differentiable_in x0 by A7, PDIFF_1:def_11;
hence  f * (reproj (i,x)) is_differentiable_in x0 by Th40, A1; ::_thesis: verum
end;
now__::_thesis:_for_z_being_set_st_z_in_].p,q.[_holds_
z_in_[.p,q.]
let z be set ; ::_thesis: ( z in ].p,q.[ implies z in [.p,q.] )
assume  z in ].p,q.[ ; ::_thesis: z in [.p,q.]
then  ex z1 being Real st
( z = z1 & p < z1 & z1 < q ) ;
hence  z in [.p,q.] ; ::_thesis: verum
end;
then A8: ].p,q.[ c= [.p,q.] by TARSKI:def_3;
then A9: ].p,q.[ c= dom (f * (reproj (i,x))) by A4, XBOOLE_1:1;
 for x being Real st x in ].p,q.[ holds
f * (reproj (i,x)) is_differentiable_in x by A5, A8;
then A10: f * (reproj (i,x)) is_differentiable_on ].p,q.[ by A9, FDIFF_1:9;
now__::_thesis:_for_x0,_r_being_real_number_st_x0_in_[.p,q.]_&_0_<_r_holds_
ex_s_being_real_number_st_
(_0_<_s_&_(_for_x1_being_real_number_st_x1_in_[.p,q.]_&_abs_(x1_-_x0)_<_s_holds_
abs_(((f_*_(reproj_(i,x)))_._x1)_-_((f_*_(reproj_(i,x)))_._x0))_<_r_)_)
let x0, r be real number ; ::_thesis: ( x0 in [.p,q.] & 0 < r implies ex s being real number st
( 0 < s & ( for x1 being real number st x1 in [.p,q.] & abs (x1 - x0) < s holds
abs (((f * (reproj (i,x))) . x1) - ((f * (reproj (i,x))) . x0)) < r ) ) )
assume A11: ( x0 in [.p,q.] & 0 < r ) ; ::_thesis: ex s being real number st
( 0 < s & ( for x1 being real number st x1 in [.p,q.] & abs (x1 - x0) < s holds
abs (((f * (reproj (i,x))) . x1) - ((f * (reproj (i,x))) . x0)) < r ) )
then  f * (reproj (i,x)) is_continuous_in x0 by A5, FDIFF_1:24;
then consider s being real number  such that
A12: ( 0 < s & ( for x1 being real number st x1 in dom (f * (reproj (i,x))) & abs (x1 - x0) < s holds
abs (((f * (reproj (i,x))) . x1) - ((f * (reproj (i,x))) . x0)) < r ) ) by A11, FCONT_1:3;
take s = s; ::_thesis: ( 0 < s & ( for x1 being real number st x1 in [.p,q.] & abs (x1 - x0) < s holds
abs (((f * (reproj (i,x))) . x1) - ((f * (reproj (i,x))) . x0)) < r ) )
thus  0 < s by A12; ::_thesis: for x1 being real number st x1 in [.p,q.] & abs (x1 - x0) < s holds
abs (((f * (reproj (i,x))) . x1) - ((f * (reproj (i,x))) . x0)) < r
thus  for x1 being real number st x1 in [.p,q.] & abs (x1 - x0) < s holds
abs (((f * (reproj (i,x))) . x1) - ((f * (reproj (i,x))) . x0)) < r by A4, A12; ::_thesis: verum
end;
then  (f * (reproj (i,x))) | [.p,q.] is continuous by A4, FCONT_1:14;
then consider r being Real such that
A13: ( r in ].p,q.[ & diff ((f * (reproj (i,x))),r) = (((f * (reproj (i,x))) . q) - ((f * (reproj (i,x))) . p)) / (q - p) ) by A1, A4, A10, ROLLE:3;
 q - p <> 0 by A1;
then A14: (diff ((f * (reproj (i,x))),r)) * (q - p) = ((f * (reproj (i,x))) . q) - ((f * (reproj (i,x))) . p) by A13, XCMPLX_1:87;
A15: p in  {  s where s is Real : ( p <= s & s <= q )  }  by A1;
then A16: f /. ((reproj (i,x)) . p) = f . ((reproj (i,x)) . p) by A1, PARTFUN1:def_6
.= (f * (reproj (i,x))) . p by A4, A15, FUNCT_1:12 ;
A17: q in  {  s where s is Real : ( p <= s & s <= q )  }  by A1;
then A18: f /. ((reproj (i,x)) . q) = f . ((reproj (i,x)) . q) by A1, PARTFUN1:def_6
.= (f * (reproj (i,x))) . q by A4, A17, FUNCT_1:12 ;
reconsider y = (reproj (i,x)) . r as Element of REAL m ;
 diff ((f * (reproj (i,x))),r) = partdiff (f,y,i) by A1, Th41;
hence  ex r being Real ex y being Element of REAL m st
( r in ].p,q.[ & y = (reproj (i,x)) . r & (f /. ((reproj (i,x)) . q)) - (f /. ((reproj (i,x)) . p)) = (q - p) * (partdiff (f,y,i)) ) by A13, A14, A16, A18; ::_thesis: verum
end;

theorem Th43: :: PDIFF_7:43
for m being non empty Element of NAT
for f being PartFunc of (REAL m),REAL
for p, q being Real
for x being Element of REAL m
for i being Element of NAT st 1 <= i & i <= m & p <= q & ( for h being Real st h in [.p,q.] holds
(reproj (i,x)) . h in dom f ) & ( for h being Real st h in [.p,q.] holds
f is_partial_differentiable_in (reproj (i,x)) . h,i ) holds
ex r being Real ex y being Element of REAL m st
( r in [.p,q.] & y = (reproj (i,x)) . r & (f /. ((reproj (i,x)) . q)) - (f /. ((reproj (i,x)) . p)) = (q - p) * (partdiff (f,y,i)) )
proof
let m be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),REAL
for p, q being Real
for x being Element of REAL m
for i being Element of NAT st 1 <= i & i <= m & p <= q & ( for h being Real st h in [.p,q.] holds
(reproj (i,x)) . h in dom f ) & ( for h being Real st h in [.p,q.] holds
f is_partial_differentiable_in (reproj (i,x)) . h,i ) holds
ex r being Real ex y being Element of REAL m st
( r in [.p,q.] & y = (reproj (i,x)) . r & (f /. ((reproj (i,x)) . q)) - (f /. ((reproj (i,x)) . p)) = (q - p) * (partdiff (f,y,i)) )
let f be PartFunc of (REAL m),REAL; ::_thesis: for p, q being Real
for x being Element of REAL m
for i being Element of NAT st 1 <= i & i <= m & p <= q & ( for h being Real st h in [.p,q.] holds
(reproj (i,x)) . h in dom f ) & ( for h being Real st h in [.p,q.] holds
f is_partial_differentiable_in (reproj (i,x)) . h,i ) holds
ex r being Real ex y being Element of REAL m st
( r in [.p,q.] & y = (reproj (i,x)) . r & (f /. ((reproj (i,x)) . q)) - (f /. ((reproj (i,x)) . p)) = (q - p) * (partdiff (f,y,i)) )
let p, q be Real; ::_thesis: for x being Element of REAL m
for i being Element of NAT st 1 <= i & i <= m & p <= q & ( for h being Real st h in [.p,q.] holds
(reproj (i,x)) . h in dom f ) & ( for h being Real st h in [.p,q.] holds
f is_partial_differentiable_in (reproj (i,x)) . h,i ) holds
ex r being Real ex y being Element of REAL m st
( r in [.p,q.] & y = (reproj (i,x)) . r & (f /. ((reproj (i,x)) . q)) - (f /. ((reproj (i,x)) . p)) = (q - p) * (partdiff (f,y,i)) )
let x be Element of REAL m; ::_thesis: for i being Element of NAT st 1 <= i & i <= m & p <= q & ( for h being Real st h in [.p,q.] holds
(reproj (i,x)) . h in dom f ) & ( for h being Real st h in [.p,q.] holds
f is_partial_differentiable_in (reproj (i,x)) . h,i ) holds
ex r being Real ex y being Element of REAL m st
( r in [.p,q.] & y = (reproj (i,x)) . r & (f /. ((reproj (i,x)) . q)) - (f /. ((reproj (i,x)) . p)) = (q - p) * (partdiff (f,y,i)) )
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m & p <= q & ( for h being Real st h in [.p,q.] holds
(reproj (i,x)) . h in dom f ) & ( for h being Real st h in [.p,q.] holds
f is_partial_differentiable_in (reproj (i,x)) . h,i ) implies ex r being Real ex y being Element of REAL m st
( r in [.p,q.] & y = (reproj (i,x)) . r & (f /. ((reproj (i,x)) . q)) - (f /. ((reproj (i,x)) . p)) = (q - p) * (partdiff (f,y,i)) ) )
assume A1: ( 1 <= i & i <= m & p <= q & ( for h being Real st h in [.p,q.] holds
(reproj (i,x)) . h in dom f ) & ( for h being Real st h in [.p,q.] holds
f is_partial_differentiable_in (reproj (i,x)) . h,i ) ) ; ::_thesis: ex r being Real ex y being Element of REAL m st
( r in [.p,q.] & y = (reproj (i,x)) . r & (f /. ((reproj (i,x)) . q)) - (f /. ((reproj (i,x)) . p)) = (q - p) * (partdiff (f,y,i)) )
percases ( p = q or p <> q ) ;
supposeA2: p = q ; ::_thesis: ex r being Real ex y being Element of REAL m st
( r in [.p,q.] & y = (reproj (i,x)) . r & (f /. ((reproj (i,x)) . q)) - (f /. ((reproj (i,x)) . p)) = (q - p) * (partdiff (f,y,i)) )
then A3: p in [.p,q.] ;
reconsider y = (reproj (i,x)) . p as Element of REAL m ;
 (q - p) * (partdiff (f,y,i)) = 0 by A2;
hence  ex r being Real ex y being Element of REAL m st
( r in [.p,q.] & y = (reproj (i,x)) . r & (f /. ((reproj (i,x)) . q)) - (f /. ((reproj (i,x)) . p)) = (q - p) * (partdiff (f,y,i)) ) by A3, A2; ::_thesis: verum
end;
suppose p <> q ; ::_thesis: ex r being Real ex y being Element of REAL m st
( r in [.p,q.] & y = (reproj (i,x)) . r & (f /. ((reproj (i,x)) . q)) - (f /. ((reproj (i,x)) . p)) = (q - p) * (partdiff (f,y,i)) )
then  p < q by A1, XXREAL_0:1;
then A4: ex r being Real ex y being Element of REAL m st
( r in ].p,q.[ & y = (reproj (i,x)) . r & (f /. ((reproj (i,x)) . q)) - (f /. ((reproj (i,x)) . p)) = (q - p) * (partdiff (f,y,i)) ) by Th42, A1;
 ].p,q.[ c= [.p,q.] by XXREAL_1:25;
hence  ex r being Real ex y being Element of REAL m st
( r in [.p,q.] & y = (reproj (i,x)) . r & (f /. ((reproj (i,x)) . q)) - (f /. ((reproj (i,x)) . p)) = (q - p) * (partdiff (f,y,i)) ) by A4; ::_thesis: verum
end;
end;
end;

theorem Th44: :: PDIFF_7:44
for m being non empty Element of NAT
for x, y, z, w being Element of REAL m
for i being Element of NAT
for d, p, q, r being Real st 1 <= i & i <= m & |.(y - x).| < d & |.(z - x).| < d & p = (proj (i,m)) . y & z = (reproj (i,y)) . q & r in [.p,q.] & w = (reproj (i,y)) . r holds
|.(w - x).| < d
proof
let m be non empty Element of NAT ; ::_thesis: for x, y, z, w being Element of REAL m
for i being Element of NAT
for d, p, q, r being Real st 1 <= i & i <= m & |.(y - x).| < d & |.(z - x).| < d & p = (proj (i,m)) . y & z = (reproj (i,y)) . q & r in [.p,q.] & w = (reproj (i,y)) . r holds
|.(w - x).| < d
let x, y, z, w be Element of REAL m; ::_thesis: for i being Element of NAT
for d, p, q, r being Real st 1 <= i & i <= m & |.(y - x).| < d & |.(z - x).| < d & p = (proj (i,m)) . y & z = (reproj (i,y)) . q & r in [.p,q.] & w = (reproj (i,y)) . r holds
|.(w - x).| < d
let i be Element of NAT ; ::_thesis: for d, p, q, r being Real st 1 <= i & i <= m & |.(y - x).| < d & |.(z - x).| < d & p = (proj (i,m)) . y & z = (reproj (i,y)) . q & r in [.p,q.] & w = (reproj (i,y)) . r holds
|.(w - x).| < d
let d, p, q, r be Real; ::_thesis: ( 1 <= i & i <= m & |.(y - x).| < d & |.(z - x).| < d & p = (proj (i,m)) . y & z = (reproj (i,y)) . q & r in [.p,q.] & w = (reproj (i,y)) . r implies |.(w - x).| < d )
assume that
A1: ( 1 <= i & i <= m ) and
A2: ( |.(y - x).| < d & |.(z - x).| < d ) and
A3: ( p = (proj (i,m)) . y & z = (reproj (i,y)) . q ) and
A4: r in [.p,q.] and
A5: w = (reproj (i,y)) . r ; ::_thesis: |.(w - x).| < d
set wx = w - x;
set yx = y - x;
set zx = z - x;
A6: ( Sum (sqr (y - x)) = |.(y - x).| ^2 & Sum (sqr (w - x)) = |.(w - x).| ^2 & Sum (sqr (z - x)) = |.(z - x).| ^2 ) by TOPREAL9:5;
A7: ( (proj (i,m)) . z = q & (proj (i,m)) . w = r ) by A1, A3, A5, Th39;
A8: ( p <= r & r <= q ) by A4, XXREAL_1:1;
 i in Seg m by A1;
then A9: ( i in dom (y - x) & i in dom (w - x) & i in dom (z - x) ) by FINSEQ_1:89;
set x1 = x;
A10: for l being Element of NAT st l in Seg m & l <> i holds
(sqr (y - x)) . l = (sqr (w - x)) . l
proof
let l be Element of NAT ; ::_thesis: ( l in Seg m & l <> i implies (sqr (y - x)) . l = (sqr (w - x)) . l )
assume A11: ( l in Seg m & l <> i ) ; ::_thesis: (sqr (y - x)) . l = (sqr (w - x)) . l
then A12: ( l in dom (y - x) & l in dom (w - x) & l in dom y ) by FINSEQ_1:89;
then A13: l in Seg (len y) by FINSEQ_1:def_3;
then A14: ( 1 <= l & l <= len y ) by FINSEQ_1:1;
 ( l in Seg (len y) & l in Seg (len (Replace (y,i,r))) ) by A13, FINSEQ_7:5;
then A15: ( l in dom y & l in dom (Replace (y,i,r)) ) by FINSEQ_1:def_3;
 ( sqr (y - x) = sqrreal * (y - x) & sqr (w - x) = sqrreal * (w - x) ) by RVSUM_1:def_8;
then  ( (sqr (y - x)) . l = sqrreal . ((y - x) . l) & (sqr (w - x)) . l = sqrreal . ((w - x) . l) ) by A12, FUNCT_1:13;
then  ( (sqr (y - x)) . l = ((y - x) . l) ^2 & (sqr (w - x)) . l = ((w - x) . l) ^2 ) by RVSUM_1:def_2;
then A16: ( (sqr (y - x)) . l = ((y . l) - (x . l)) ^2 & (sqr (w - x)) . l = ((w . l) - (x . l)) ^2 ) by A12, VALUED_1:13;
 w . l = (Replace (y,i,r)) . l by A5, PDIFF_1:def_5;
then  w . l = (Replace (y,i,r)) /. l by A15, PARTFUN1:def_6;
then  w . l = y /. l by A11, A14, FINSEQ_7:10;
hence  (sqr (y - x)) . l = (sqr (w - x)) . l by A15, A16, PARTFUN1:def_6; ::_thesis: verum
end;
A17: for l being Element of NAT st l in Seg m & l <> i holds
(sqr (z - x)) . l = (sqr (w - x)) . l
proof
let l be Element of NAT ; ::_thesis: ( l in Seg m & l <> i implies (sqr (z - x)) . l = (sqr (w - x)) . l )
assume A18: ( l in Seg m & l <> i ) ; ::_thesis: (sqr (z - x)) . l = (sqr (w - x)) . l
then A19: ( l in dom (z - x) & l in dom (w - x) & l in dom z ) by FINSEQ_1:89;
then A20: l in Seg (len z) by FINSEQ_1:def_3;
then A21: ( 1 <= l & l <= len z ) by FINSEQ_1:1;
 ( l in Seg (len z) & l in Seg (len (Replace (z,i,r))) ) by A20, FINSEQ_7:5;
then A22: ( l in dom z & l in dom (Replace (z,i,r)) ) by FINSEQ_1:def_3;
 ( sqr (z - x) = sqrreal * (z - x) & sqr (w - x) = sqrreal * (w - x) ) by RVSUM_1:def_8;
then  ( (sqr (z - x)) . l = sqrreal . ((z - x) . l) & (sqr (w - x)) . l = sqrreal . ((w - x) . l) ) by A19, FUNCT_1:13;
then  ( (sqr (z - x)) . l = ((z - x) . l) ^2 & (sqr (w - x)) . l = ((w - x) . l) ^2 ) by RVSUM_1:def_2;
then A23: ( (sqr (z - x)) . l = ((z . l) - (x . l)) ^2 & (sqr (w - x)) . l = ((w . l) - (x . l)) ^2 ) by A19, VALUED_1:13;
 w . l = ((reproj (i,z)) . r) . l by A1, A3, Th40, A5;
then  w . l = (Replace (z,i,r)) . l by PDIFF_1:def_5;
then  w . l = (Replace (z,i,r)) /. l by A22, PARTFUN1:def_6;
then  w . l = z /. l by A18, A21, FINSEQ_7:10;
hence  (sqr (z - x)) . l = (sqr (w - x)) . l by A22, A23, PARTFUN1:def_6; ::_thesis: verum
end;
A24: now__::_thesis:_(_|.(w_-_x).|_>_|.(y_-_x).|_implies_not_|.(w_-_x).|_>_|.(z_-_x).|_)
assume A25: ( |.(w - x).| > |.(y - x).| & |.(w - x).| > |.(z - x).| ) ; ::_thesis: contradiction
A26: now__::_thesis:_not_(sqr_(w_-_x))_._i_<=_(sqr_(y_-_x))_._i
assume A27: (sqr (w - x)) . i <= (sqr (y - x)) . i ; ::_thesis: contradiction
A28: len (sqr (w - x)) = m by CARD_1:def_7
.= len (sqr (y - x)) by CARD_1:def_7 ;
 for l being Element of NAT st l in dom (sqr (w - x)) holds
(sqr (w - x)) . l <= (sqr (y - x)) . l
proof
let l be Element of NAT ; ::_thesis: ( l in dom (sqr (w - x)) implies (sqr (w - x)) . l <= (sqr (y - x)) . l )
assume  l in dom (sqr (w - x)) ; ::_thesis: (sqr (w - x)) . l <= (sqr (y - x)) . l
then A29: l in Seg m by FINSEQ_1:89;
percases ( l = i or l <> i ) ;
suppose l = i ; ::_thesis: (sqr (w - x)) . l <= (sqr (y - x)) . l
hence  (sqr (w - x)) . l <= (sqr (y - x)) . l by A27; ::_thesis: verum
end;
suppose l <> i ; ::_thesis: (sqr (w - x)) . l <= (sqr (y - x)) . l
hence  (sqr (w - x)) . l <= (sqr (y - x)) . l by A29, A10; ::_thesis: verum
end;
end;
end;
hence  contradiction by A28, A6, A25, INTEGRA5:3, SQUARE_1:16; ::_thesis: verum
end;
A30: now__::_thesis:_not_(sqr_(w_-_x))_._i_<=_(sqr_(z_-_x))_._i
assume A31: (sqr (w - x)) . i <= (sqr (z - x)) . i ; ::_thesis: contradiction
A32: len (sqr (w - x)) = m by CARD_1:def_7
.= len (sqr (z - x)) by CARD_1:def_7 ;
 for l being Element of NAT st l in dom (sqr (w - x)) holds
(sqr (w - x)) . l <= (sqr (z - x)) . l
proof
let l be Element of NAT ; ::_thesis: ( l in dom (sqr (w - x)) implies (sqr (w - x)) . l <= (sqr (z - x)) . l )
assume  l in dom (sqr (w - x)) ; ::_thesis: (sqr (w - x)) . l <= (sqr (z - x)) . l
then A33: l in Seg m by FINSEQ_1:89;
percases ( l = i or l <> i ) ;
suppose l = i ; ::_thesis: (sqr (w - x)) . l <= (sqr (z - x)) . l
hence  (sqr (w - x)) . l <= (sqr (z - x)) . l by A31; ::_thesis: verum
end;
suppose l <> i ; ::_thesis: (sqr (w - x)) . l <= (sqr (z - x)) . l
hence  (sqr (w - x)) . l <= (sqr (z - x)) . l by A33, A17; ::_thesis: verum
end;
end;
end;
hence  contradiction by A32, A6, A25, INTEGRA5:3, SQUARE_1:16; ::_thesis: verum
end;
 ( sqr (y - x) = sqrreal * (y - x) & sqr (w - x) = sqrreal * (w - x) & sqr (z - x) = sqrreal * (z - x) ) by RVSUM_1:def_8;
then  ( (sqr (y - x)) . i = sqrreal . ((y - x) . i) & (sqr (w - x)) . i = sqrreal . ((w - x) . i) & (sqr (z - x)) . i = sqrreal . ((z - x) . i) ) by A9, FUNCT_1:13;
then A34: ( (sqr (y - x)) . i = ((y - x) . i) ^2 & (sqr (w - x)) . i = ((w - x) . i) ^2 & (sqr (z - x)) . i = ((z - x) . i) ^2 ) by RVSUM_1:def_2;
 ( y . i = p & w . i = r & z . i = q ) by A3, A7, PDIFF_1:def_1;
then A35: ( (sqr (y - x)) . i = (p - (x . i)) ^2 & (sqr (w - x)) . i = (r - (x . i)) ^2 & (sqr (z - x)) . i = (q - (x . i)) ^2 ) by A34, A9, VALUED_1:13;
A36: p <= q by A8, XXREAL_0:2;
percases ( x . i < p or ( p <= x . i & x . i <= r ) or ( r < x . i & x . i <= q ) or q < x . i ) ;
suppose x . i < p ; ::_thesis: contradiction
then  ( x . i < r & x . i < q ) by A8, A36, XXREAL_0:2;
then  ( q - (x . i) > 0 & r - (x . i) > 0 ) by XREAL_1:50;
then  q - (x . i) < r - (x . i) by A35, A30, SQUARE_1:15;
hence  contradiction by A8, XREAL_1:13; ::_thesis: verum
end;
supposeA37: ( p <= x . i & x . i <= r ) ; ::_thesis: contradiction
then  x . i <= q by A8, XXREAL_0:2;
then  ( r - (x . i) >= 0 & q - (x . i) >= 0 ) by A37, XREAL_1:48;
then  q - (x . i) < r - (x . i) by A35, A30, SQUARE_1:15;
hence  contradiction by A8, XREAL_1:13; ::_thesis: verum
end;
supposeA38: ( r < x . i & x . i <= q ) ; ::_thesis: contradiction
then  p < x . i by A8, XXREAL_0:2;
then A39: ( (x . i) - p >= 0 & (x . i) - r >= 0 ) by A38, XREAL_1:48;
 ( (p - (x . i)) ^2 = ((x . i) - p) ^2 & (r - (x . i)) ^2 = ((x . i) - r) ^2 ) ;
then  (x . i) - p < (x . i) - r by A35, A26, A39, SQUARE_1:15;
hence  contradiction by A8, XREAL_1:13; ::_thesis: verum
end;
suppose q < x . i ; ::_thesis: contradiction
then  r < x . i by A8, XXREAL_0:2;
then  ( p < x . i & r < x . i ) by A8, XXREAL_0:2;
then A40: ( (x . i) - r >= 0 & (x . i) - p >= 0 ) by XREAL_1:48;
 ( (p - (x . i)) ^2 = ((x . i) - p) ^2 & (r - (x . i)) ^2 = ((x . i) - r) ^2 ) ;
then  (x . i) - p < (x . i) - r by A35, A26, A40, SQUARE_1:15;
hence  contradiction by A8, XREAL_1:13; ::_thesis: verum
end;
end;
end;
percases ( |.(w - x).| <= |.(y - x).| or |.(w - x).| <= |.(z - x).| ) by A24;
suppose |.(w - x).| <= |.(y - x).| ; ::_thesis: |.(w - x).| < d
hence  |.(w - x).| < d by A2, XXREAL_0:2; ::_thesis: verum
end;
suppose |.(w - x).| <= |.(z - x).| ; ::_thesis: |.(w - x).| < d
hence  |.(w - x).| < d by A2, XXREAL_0:2; ::_thesis: verum
end;
end;
end;

theorem Th45: :: PDIFF_7:45
for m being non empty Element of NAT
for f being PartFunc of (REAL m),REAL
for X being Subset of (REAL m)
for x, y, z being Element of REAL m
for i being Element of NAT
for d, p, q being Real st 1 <= i & i <= m & X is open & x in X & |.(y - x).| < d & |.(z - x).| < d & X c= dom f & ( for x being Element of REAL m st x in X holds
f is_partial_differentiable_in x,i ) & 0 < d & ( for z being Element of REAL m st |.(z - x).| < d holds
z in X ) & z = (reproj (i,y)) . p & q = (proj (i,m)) . y holds
ex w being Element of REAL m st
( |.(w - x).| < d & f is_partial_differentiable_in w,i & (f /. z) - (f /. y) = (p - q) * (partdiff (f,w,i)) )
proof
let m be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),REAL
for X being Subset of (REAL m)
for x, y, z being Element of REAL m
for i being Element of NAT
for d, p, q being Real st 1 <= i & i <= m & X is open & x in X & |.(y - x).| < d & |.(z - x).| < d & X c= dom f & ( for x being Element of REAL m st x in X holds
f is_partial_differentiable_in x,i ) & 0 < d & ( for z being Element of REAL m st |.(z - x).| < d holds
z in X ) & z = (reproj (i,y)) . p & q = (proj (i,m)) . y holds
ex w being Element of REAL m st
( |.(w - x).| < d & f is_partial_differentiable_in w,i & (f /. z) - (f /. y) = (p - q) * (partdiff (f,w,i)) )
let f be PartFunc of (REAL m),REAL; ::_thesis: for X being Subset of (REAL m)
for x, y, z being Element of REAL m
for i being Element of NAT
for d, p, q being Real st 1 <= i & i <= m & X is open & x in X & |.(y - x).| < d & |.(z - x).| < d & X c= dom f & ( for x being Element of REAL m st x in X holds
f is_partial_differentiable_in x,i ) & 0 < d & ( for z being Element of REAL m st |.(z - x).| < d holds
z in X ) & z = (reproj (i,y)) . p & q = (proj (i,m)) . y holds
ex w being Element of REAL m st
( |.(w - x).| < d & f is_partial_differentiable_in w,i & (f /. z) - (f /. y) = (p - q) * (partdiff (f,w,i)) )
let X be Subset of (REAL m); ::_thesis: for x, y, z being Element of REAL m
for i being Element of NAT
for d, p, q being Real st 1 <= i & i <= m & X is open & x in X & |.(y - x).| < d & |.(z - x).| < d & X c= dom f & ( for x being Element of REAL m st x in X holds
f is_partial_differentiable_in x,i ) & 0 < d & ( for z being Element of REAL m st |.(z - x).| < d holds
z in X ) & z = (reproj (i,y)) . p & q = (proj (i,m)) . y holds
ex w being Element of REAL m st
( |.(w - x).| < d & f is_partial_differentiable_in w,i & (f /. z) - (f /. y) = (p - q) * (partdiff (f,w,i)) )
let x, y, z be Element of REAL m; ::_thesis: for i being Element of NAT
for d, p, q being Real st 1 <= i & i <= m & X is open & x in X & |.(y - x).| < d & |.(z - x).| < d & X c= dom f & ( for x being Element of REAL m st x in X holds
f is_partial_differentiable_in x,i ) & 0 < d & ( for z being Element of REAL m st |.(z - x).| < d holds
z in X ) & z = (reproj (i,y)) . p & q = (proj (i,m)) . y holds
ex w being Element of REAL m st
( |.(w - x).| < d & f is_partial_differentiable_in w,i & (f /. z) - (f /. y) = (p - q) * (partdiff (f,w,i)) )
let i be Element of NAT ; ::_thesis: for d, p, q being Real st 1 <= i & i <= m & X is open & x in X & |.(y - x).| < d & |.(z - x).| < d & X c= dom f & ( for x being Element of REAL m st x in X holds
f is_partial_differentiable_in x,i ) & 0 < d & ( for z being Element of REAL m st |.(z - x).| < d holds
z in X ) & z = (reproj (i,y)) . p & q = (proj (i,m)) . y holds
ex w being Element of REAL m st
( |.(w - x).| < d & f is_partial_differentiable_in w,i & (f /. z) - (f /. y) = (p - q) * (partdiff (f,w,i)) )
let d, p, q be Real; ::_thesis: ( 1 <= i & i <= m & X is open & x in X & |.(y - x).| < d & |.(z - x).| < d & X c= dom f & ( for x being Element of REAL m st x in X holds
f is_partial_differentiable_in x,i ) & 0 < d & ( for z being Element of REAL m st |.(z - x).| < d holds
z in X ) & z = (reproj (i,y)) . p & q = (proj (i,m)) . y implies ex w being Element of REAL m st
( |.(w - x).| < d & f is_partial_differentiable_in w,i & (f /. z) - (f /. y) = (p - q) * (partdiff (f,w,i)) ) )
assume A1: ( 1 <= i & i <= m & X is open & x in X & |.(y - x).| < d & |.(z - x).| < d & X c= dom f & ( for x being Element of REAL m st x in X holds
f is_partial_differentiable_in x,i ) & 0 < d & ( for z being Element of REAL m st |.(z - x).| < d holds
z in X ) & z = (reproj (i,y)) . p & q = (proj (i,m)) . y ) ; ::_thesis: ex w being Element of REAL m st
( |.(w - x).| < d & f is_partial_differentiable_in w,i & (f /. z) - (f /. y) = (p - q) * (partdiff (f,w,i)) )
then A2: p = (proj (i,m)) . z by Th39;
 (reproj (i,y)) . q = (reproj (i,y)) . (y . i) by A1, PDIFF_1:def_1;
then  (reproj (i,y)) . q = Replace (y,i,(y . i)) by PDIFF_1:def_5;
then A3: y = (reproj (i,y)) . q by FUNCT_7:35;
percases ( q <= p or p < q ) ;
supposeA4: q <= p ; ::_thesis: ex w being Element of REAL m st
( |.(w - x).| < d & f is_partial_differentiable_in w,i & (f /. z) - (f /. y) = (p - q) * (partdiff (f,w,i)) )
A5: for h being Real st h in [.q,p.] holds
(reproj (i,y)) . h in X
proof
let h be Real; ::_thesis: ( h in [.q,p.] implies (reproj (i,y)) . h in X )
assume  h in [.q,p.] ; ::_thesis: (reproj (i,y)) . h in X
then  |.(((reproj (i,y)) . h) - x).| < d by A1, Th44;
hence  (reproj (i,y)) . h in X by A1; ::_thesis: verum
end;
A6: now__::_thesis:_for_h_being_Real_st_h_in_[.q,p.]_holds_
(reproj_(i,y))_._h_in_dom_f
let h be Real; ::_thesis: ( h in [.q,p.] implies (reproj (i,y)) . h in dom f )
assume  h in [.q,p.] ; ::_thesis: (reproj (i,y)) . h in dom f
then  (reproj (i,y)) . h in X by A5;
hence  (reproj (i,y)) . h in dom f by A1; ::_thesis: verum
end;
now__::_thesis:_for_h_being_Real_st_h_in_[.q,p.]_holds_
f_is_partial_differentiable_in_(reproj_(i,y))_._h,i
let h be Real; ::_thesis: ( h in [.q,p.] implies f is_partial_differentiable_in (reproj (i,y)) . h,i )
assume  h in [.q,p.] ; ::_thesis: f is_partial_differentiable_in (reproj (i,y)) . h,i
then  (reproj (i,y)) . h in X by A5;
hence  f is_partial_differentiable_in (reproj (i,y)) . h,i by A1; ::_thesis: verum
end;
then consider r being Real, w being Element of REAL m such that
A7: ( r in [.q,p.] & w = (reproj (i,y)) . r & (f /. ((reproj (i,y)) . p)) - (f /. ((reproj (i,y)) . q)) = (p - q) * (partdiff (f,w,i)) ) by Th43, A1, A4, A6;
A8: |.(w - x).| < d by A7, A1, Th44;
then  f is_partial_differentiable_in w,i by A1;
hence  ex w being Element of REAL m st
( |.(w - x).| < d & f is_partial_differentiable_in w,i & (f /. z) - (f /. y) = (p - q) * (partdiff (f,w,i)) ) by A7, A3, A1, A8; ::_thesis: verum
end;
supposeA9: p < q ; ::_thesis: ex w being Element of REAL m st
( |.(w - x).| < d & f is_partial_differentiable_in w,i & (f /. z) - (f /. y) = (p - q) * (partdiff (f,w,i)) )
A10: dom y = Seg m by FINSEQ_1:89;
then A11: ( i in dom y & len y = m ) by A1, FINSEQ_1:def_3;
then  len (Replace (y,i,p)) = m by FINSEQ_7:5;
then A12: dom y = dom (Replace (y,i,p)) by A10, FINSEQ_1:def_3;
 z = Replace (y,i,p) by A1, PDIFF_1:def_5;
then  z . i = (Replace (y,i,p)) /. i by A11, A12, PARTFUN1:def_6;
then A13: z . i = p by A1, A11, FINSEQ_7:8;
A14: y = (reproj (i,z)) . q by A1, Th40, A3;
A15: for h being Real st h in [.p,q.] holds
(reproj (i,z)) . h in X
proof
let h be Real; ::_thesis: ( h in [.p,q.] implies (reproj (i,z)) . h in X )
assume  h in [.p,q.] ; ::_thesis: (reproj (i,z)) . h in X
then  |.(((reproj (i,z)) . h) - x).| < d by A2, A14, A1, Th44;
hence  (reproj (i,z)) . h in X by A1; ::_thesis: verum
end;
A16: for h being Real st h in [.p,q.] holds
(reproj (i,z)) . h in dom f
proof
let h be Real; ::_thesis: ( h in [.p,q.] implies (reproj (i,z)) . h in dom f )
assume  h in [.p,q.] ; ::_thesis: (reproj (i,z)) . h in dom f
then  (reproj (i,z)) . h in X by A15;
hence  (reproj (i,z)) . h in dom f by A1; ::_thesis: verum
end;
 for h being Real st h in [.p,q.] holds
f is_partial_differentiable_in (reproj (i,z)) . h,i
proof
let h be Real; ::_thesis: ( h in [.p,q.] implies f is_partial_differentiable_in (reproj (i,z)) . h,i )
assume  h in [.p,q.] ; ::_thesis: f is_partial_differentiable_in (reproj (i,z)) . h,i
then  (reproj (i,z)) . h in X by A15;
hence  f is_partial_differentiable_in (reproj (i,z)) . h,i by A1; ::_thesis: verum
end;
then consider r being Real, w being Element of REAL m such that
A17: ( r in [.p,q.] & w = (reproj (i,z)) . r & (f /. ((reproj (i,z)) . q)) - (f /. ((reproj (i,z)) . p)) = (q - p) * (partdiff (f,w,i)) ) by Th43, A1, A9, A16;
A18: |.(w - x).| < d by A2, A14, A17, A1, Th44;
then A19: f is_partial_differentiable_in w,i by A1;
 (reproj (i,z)) . p = Replace (z,i,(z . i)) by A13, PDIFF_1:def_5;
then  (f /. y) - (f /. z) = (q - p) * (partdiff (f,w,i)) by A14, A17, FUNCT_7:35;
then  (f /. z) - (f /. y) = (p - q) * (partdiff (f,w,i)) ;
hence  ex w being Element of REAL m st
( |.(w - x).| < d & f is_partial_differentiable_in w,i & (f /. z) - (f /. y) = (p - q) * (partdiff (f,w,i)) ) by A18, A19; ::_thesis: verum
end;
end;
end;

theorem Th46: :: PDIFF_7:46
for m being non empty Element of NAT
for h being FinSequence of REAL m
for y, x being Element of REAL m
for j being Element of NAT st len h = m + 1 & 1 <= j & j <= m & ( for i being Nat st i in dom h holds
h /. i = (y | ((m + 1) -' i)) ^ (0* (i -' 1)) ) holds
x + (h /. j) = (reproj (((m + 1) -' j),(x + (h /. (j + 1))))) . ((proj (((m + 1) -' j),m)) . (x + y))
proof
let m be non empty Element of NAT ; ::_thesis: for h being FinSequence of REAL m
for y, x being Element of REAL m
for j being Element of NAT st len h = m + 1 & 1 <= j & j <= m & ( for i being Nat st i in dom h holds
h /. i = (y | ((m + 1) -' i)) ^ (0* (i -' 1)) ) holds
x + (h /. j) = (reproj (((m + 1) -' j),(x + (h /. (j + 1))))) . ((proj (((m + 1) -' j),m)) . (x + y))
let h be FinSequence of REAL m; ::_thesis: for y, x being Element of REAL m
for j being Element of NAT st len h = m + 1 & 1 <= j & j <= m & ( for i being Nat st i in dom h holds
h /. i = (y | ((m + 1) -' i)) ^ (0* (i -' 1)) ) holds
x + (h /. j) = (reproj (((m + 1) -' j),(x + (h /. (j + 1))))) . ((proj (((m + 1) -' j),m)) . (x + y))
let y, x be Element of REAL m; ::_thesis: for j being Element of NAT st len h = m + 1 & 1 <= j & j <= m & ( for i being Nat st i in dom h holds
h /. i = (y | ((m + 1) -' i)) ^ (0* (i -' 1)) ) holds
x + (h /. j) = (reproj (((m + 1) -' j),(x + (h /. (j + 1))))) . ((proj (((m + 1) -' j),m)) . (x + y))
let j be Element of NAT ; ::_thesis: ( len h = m + 1 & 1 <= j & j <= m & ( for i being Nat st i in dom h holds
h /. i = (y | ((m + 1) -' i)) ^ (0* (i -' 1)) ) implies x + (h /. j) = (reproj (((m + 1) -' j),(x + (h /. (j + 1))))) . ((proj (((m + 1) -' j),m)) . (x + y)) )
assume A1: ( len h = m + 1 & 1 <= j & j <= m & ( for i being Nat st i in dom h holds
h /. i = (y | ((m + 1) -' i)) ^ (0* (i -' 1)) ) ) ; ::_thesis: x + (h /. j) = (reproj (((m + 1) -' j),(x + (h /. (j + 1))))) . ((proj (((m + 1) -' j),m)) . (x + y))
reconsider mj = m - j as Element of NAT by A1, NAT_1:21;
 m <= m + 1 by NAT_1:11;
then A2: Seg m c= Seg (m + 1) by FINSEQ_1:5;
A3: j in Seg m by A1;
then  j in Seg (m + 1) by A2;
then  j in dom h by A1, FINSEQ_1:def_3;
then A4: h /. j = (y | ((m + 1) -' j)) ^ (0* (j -' 1)) by A1;
 j + 1 in Seg (m + 1) by A3, FINSEQ_1:60;
then  j + 1 in dom h by A1, FINSEQ_1:def_3;
then A5: h /. (j + 1) = (y | ((m + 1) -' (j + 1))) ^ (0* ((j + 1) -' 1)) by A1;
 (m + 1) -' j = (m -' j) + 1 by A1, NAT_D:38;
then A6: 1 <= (m + 1) -' j by NAT_1:11;
A7: 1 - j <= 0 by A1, XREAL_1:47;
(m + 1) -' j = (m + 1) - j by A1, NAT_D:37
.= m + (1 - j) ;
then A8: (m + 1) -' j <= m by A7, XREAL_1:32;
then  (m + 1) -' j in Seg m by A6;
then A9: ( (m + 1) -' j in dom (x + y) & (m + 1) -' j in dom y & (m + 1) -' j in dom x ) by FINSEQ_1:89;
 (m + 1) -' j <= len y by A8, CARD_1:def_7;
then A10: len (y | ((m + 1) -' j)) = (m + 1) -' j by FINSEQ_1:59;
 j + 1 <= m + 1 by A1, XREAL_1:6;
then A11: (m + 1) -' (j + 1) = (m + 1) - (j + 1) by XREAL_1:233;
then  ( (m + 1) -' (j + 1) = m - j & j >= 0 ) ;
then  (m + 1) -' (j + 1) <= m by XREAL_1:43;
then  (m + 1) -' (j + 1) <= len y by CARD_1:def_7;
then A12: len (y | ((m + 1) -' (j + 1))) = (m + 1) -' (j + 1) by FINSEQ_1:59;
(proj (((m + 1) -' j),m)) . (x + y) = (x + y) . ((m + 1) -' j) by PDIFF_1:def_1
.= (x . ((m + 1) -' j)) + (y . ((m + 1) -' j)) by A9, VALUED_1:def_1
.= (x . ((m + 1) -' j)) + (y /. ((m + 1) -' j)) by A9, PARTFUN1:def_6
.= (x /. ((m + 1) -' j)) + (y /. ((m + 1) -' j)) by A9, PARTFUN1:def_6 ;
then A13: (reproj (((m + 1) -' j),(x + (h /. (j + 1))))) . ((proj (((m + 1) -' j),m)) . (x + y)) = Replace ((x + (h /. (j + 1))),((m + 1) -' j),((x /. ((m + 1) -' j)) + (y /. ((m + 1) -' j)))) by PDIFF_1:def_5;
 (reproj (((m + 1) -' j),(x + (h /. (j + 1))))) /. ((proj (((m + 1) -' j),m)) . (x + y)) is Element of REAL m ;
then reconsider rep = (reproj (((m + 1) -' j),(x + (h /. (j + 1))))) /. ((proj (((m + 1) -' j),m)) . (x + y)) as FinSequence of REAL ;
reconsider hj = h /. j as Element of REAL m ;
reconsider hj1 = h /. (j + 1) as Element of REAL m ;
A14: len (x + hj) = m by CARD_1:def_7
.= len rep by CARD_1:def_7 ;
now__::_thesis:_for_n_being_Nat_st_1_<=_n_&_n_<=_len_rep_holds_
(x_+_(h_/._j))_._n_=_rep_._n
let n be Nat; ::_thesis: ( 1 <= n & n <= len rep implies (x + (h /. j)) . b1 = rep . b1 )
assume A15: ( 1 <= n & n <= len rep ) ; ::_thesis: (x + (h /. j)) . b1 = rep . b1
then A16: ( 1 <= n & n <= m ) by CARD_1:def_7;
then  n in Seg m by FINSEQ_1:1;
then A17: ( n in Seg (len (x + (h /. j))) & n in Seg (len (x + (h /. (j + 1)))) & n in Seg (len x) & n in Seg (len y) ) by CARD_1:def_7;
then A18: ( n in dom (x + (h /. j)) & n in dom (x + (h /. (j + 1))) & n in dom rep & n in dom x & n in dom y ) by A14, FINSEQ_1:def_3;
then A19: ( (x + (h /. j)) . n = (x . n) + (hj . n) & (x + (h /. (j + 1))) . n = (x . n) + (hj1 . n) ) by VALUED_1:def_1;
percases ( n < (m + 1) -' j or n = (m + 1) -' j or n > (m + 1) -' j ) by XXREAL_0:1;
supposeA20: n < (m + 1) -' j ; ::_thesis: (x + (h /. j)) . b1 = rep . b1
then A21: n in Seg ((m + 1) -' j) by A15, FINSEQ_1:1;
A22: hj . n = (y | (Seg ((m + 1) -' j))) . n by A4, A10, A15, A20, FINSEQ_1:64
.= y . n by A21, FUNCT_1:49 ;
 m <= m + 1 by NAT_1:11;
then  n < (m + 1) - j by A1, A20, XREAL_1:233, XXREAL_0:2;
then  n < mj + 1 ;
then A23: n <= mj by NAT_1:13;
then A24: n in Seg mj by A15, FINSEQ_1:1;
A25: hj1 . n = (y | (Seg mj)) . n by A11, A23, A12, A5, A15, FINSEQ_1:64
.= y . n by A24, FUNCT_1:49 ;
 ( n <> (m + 1) -' j & n <= len (x + (h /. (j + 1))) ) by A20, A17, FINSEQ_1:1;
then  rep /. n = (x + (h /. (j + 1))) /. n by A13, A15, FINSEQ_7:10;
then rep . n = (x + (h /. (j + 1))) /. n by A18, PARTFUN1:def_6
.= (x + (h /. (j + 1))) . n by A18, PARTFUN1:def_6 ;
hence  (x + (h /. j)) . n = rep . n by A19, A25, A22; ::_thesis: verum
end;
supposeA26: n = (m + 1) -' j ; ::_thesis: (x + (h /. j)) . b1 = rep . b1
then A27: n in Seg ((m + 1) -' j) by A15;
A28: hj . n = (y | (Seg ((m + 1) -' j))) . n by A4, A10, A15, A26, FINSEQ_1:64
.= y . n by A27, FUNCT_1:49 ;
 n <= len (x + (h /. (j + 1))) by A17, FINSEQ_1:1;
then  rep /. n = (x /. n) + (y /. n) by A26, A13, A15, FINSEQ_7:8;
then A29: rep . n = (x /. n) + (y /. n) by A18, PARTFUN1:def_6;
thus (x + (h /. j)) . n = (x /. n) + (y . n) by A18, A19, A28, PARTFUN1:def_6
.= rep . n by A29, A18, PARTFUN1:def_6 ; ::_thesis: verum
end;
supposeA30: n > (m + 1) -' j ; ::_thesis: (x + (h /. j)) . b1 = rep . b1
then reconsider nm = n - ((m + 1) -' j) as Element of NAT by NAT_1:21;
A32: m <= m + 1 by NAT_1:11;
 n <= len hj by A16, CARD_1:def_7;
then A33: hj . n = (0* (j -' 1)) . nm by A4, A10, A30, FINSEQ_1:24
.= 0 ;
A34: len y = m by CARD_1:def_7;
 j + 1 <= m + 1 by A1, XREAL_1:6;
then (m + 1) -' (j + 1) = (m + 1) - (j + 1) by XREAL_1:233
.= m - j
.= m -' j by A1, XREAL_1:233 ;
then A35: len (y | ((m + 1) -' (j + 1))) = m -' j by A34, FINSEQ_1:59, NAT_D:35;
 n > (m + 1) - j by A30, A32, A1, XREAL_1:233, XXREAL_0:2;
then  ( n > (m - j) + 1 & (m - j) + 1 > (m - j) + 0 ) by XREAL_1:8;
then  n > m - j by XXREAL_0:2;
then A36: n > m -' j by A1, XREAL_1:233;
then reconsider nmj = n - (m -' j) as Element of NAT by NAT_1:21;
 n <= len hj1 by A16, CARD_1:def_7;
then A39: hj1 . n = (0* ((j + 1) -' 1)) . (n - (m -' j)) by A5, A35, A36, FINSEQ_1:24
.= 0 ;
 ( n <> (m + 1) -' j & n <= len (x + (h /. (j + 1))) ) by A30, A17, FINSEQ_1:1;
then  rep /. n = (x + (h /. (j + 1))) /. n by A13, A15, FINSEQ_7:10;
then rep . n = (x + (h /. (j + 1))) /. n by A18, PARTFUN1:def_6
.= (x + (h /. (j + 1))) . n by A18, PARTFUN1:def_6 ;
hence  (x + (h /. j)) . n = rep . n by A19, A39, A33; ::_thesis: verum
end;
end;
end;
hence  x + (h /. j) = (reproj (((m + 1) -' j),(x + (h /. (j + 1))))) . ((proj (((m + 1) -' j),m)) . (x + y)) by A14, FINSEQ_1:def_17; ::_thesis: verum
end;

theorem Th47: :: PDIFF_7:47
for m being non empty Element of NAT
for f being PartFunc of (REAL m),(REAL 1)
for X being Subset of (REAL m)
for x being Element of REAL m st X is open & x in X & ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) holds
( f is_differentiable_in x & ( for h being Element of REAL m ex w being FinSequence of REAL 1 st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = ((proj (i,m)) . h) * (partdiff (f,x,i)) ) & (diff (f,x)) . h = Sum w ) ) )
proof
let m be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL 1)
for X being Subset of (REAL m)
for x being Element of REAL m st X is open & x in X & ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) holds
( f is_differentiable_in x & ( for h being Element of REAL m ex w being FinSequence of REAL 1 st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = ((proj (i,m)) . h) * (partdiff (f,x,i)) ) & (diff (f,x)) . h = Sum w ) ) )
let f be PartFunc of (REAL m),(REAL 1); ::_thesis: for X being Subset of (REAL m)
for x being Element of REAL m st X is open & x in X & ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) holds
( f is_differentiable_in x & ( for h being Element of REAL m ex w being FinSequence of REAL 1 st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = ((proj (i,m)) . h) * (partdiff (f,x,i)) ) & (diff (f,x)) . h = Sum w ) ) )
let X be Subset of (REAL m); ::_thesis: for x being Element of REAL m st X is open & x in X & ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) holds
( f is_differentiable_in x & ( for h being Element of REAL m ex w being FinSequence of REAL 1 st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = ((proj (i,m)) . h) * (partdiff (f,x,i)) ) & (diff (f,x)) . h = Sum w ) ) )
let x be Element of REAL m; ::_thesis: ( X is open & x in X & ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) implies ( f is_differentiable_in x & ( for h being Element of REAL m ex w being FinSequence of REAL 1 st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = ((proj (i,m)) . h) * (partdiff (f,x,i)) ) & (diff (f,x)) . h = Sum w ) ) ) )
assume A1: ( X is open & x in X & ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) ) ; ::_thesis: ( f is_differentiable_in x & ( for h being Element of REAL m ex w being FinSequence of REAL 1 st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = ((proj (i,m)) . h) * (partdiff (f,x,i)) ) & (diff (f,x)) . h = Sum w ) ) )
consider L being Lipschitzian LinearOperator of m,1 such that
A2: for h being Element of REAL m ex w being FinSequence of REAL 1 st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = ((proj (i,m)) . h) * (partdiff (f,x,i)) ) & L . h = Sum w ) by Lm8;
consider d0 being Real such that
A3: d0 > 0 and
A4:  {  y where y is Element of REAL m : |.(y - x).| < d0  }  c= X by A1, Th31;
set N =  {  y where y is Element of REAL m : |.(y - x).| < d0  }  ;
 1 <= m by NAT_1:14;
then  f is_partial_differentiable_on X,m by A1;
then  X c= dom f by Def4;
then A5:  {  y where y is Element of REAL m : |.(y - x).| < d0  }  c= dom f by A4, XBOOLE_1:1;
deffunc H1( Element of REAL m) ->  Element of REAL 1 = ((f /. (x + $1)) - (f /. x)) - (L . $1);
consider R being Function of (REAL m),(REAL 1) such that
A6: for h being Element of REAL m holds R . h = H1(h) from FUNCT_2:sch_4();
consider f0 being PartFunc of (REAL m),REAL such that
A7: f = <>* f0 by Th29;
A8: now__::_thesis:_for_r0_being_Real_st_r0_>_0_holds_
ex_d_being_Element_of_REAL_st_
(_0_<_d_&_(_for_y_being_Element_of_REAL_m
for_z_being_Element_of_REAL_1_st_y_<>_0*_m_&_|.y.|_<_d_&_z_=_R_._y_holds_
(|.y.|_")_*_|.z.|_<_r0_)_)
let r0 be Real; ::_thesis: ( r0 > 0 implies ex d being Element of REAL st
( 0 < d & ( for y being Element of REAL m
for z being Element of REAL 1 st y <> 0* m & |.y.| < d & z = R . y holds
(|.y.| ") * |.z.| < r0 ) ) )
assume A9: r0 > 0 ; ::_thesis: ex d being Element of REAL st
( 0 < d & ( for y being Element of REAL m
for z being Element of REAL 1 st y <> 0* m & |.y.| < d & z = R . y holds
(|.y.| ") * |.z.| < r0 ) )
set r1 = r0 / 2;
set r = (r0 / 2) / m;
defpred S1[ Nat, Element of REAL ] means  ex k being Element of NAT st
( $1 = k & 0 < $2 & ( for q being Element of REAL m st q in X & |.(q - x).| < $2 holds
|.((partdiff (f,q,k)) - (partdiff (f,x,k))).| < (r0 / 2) / m ) );
A10: for k0 being Nat st k0 in Seg m holds
ex d being Element of REAL st S1[k0,d]
proof
let k0 be Nat; ::_thesis: ( k0 in Seg m implies ex d being Element of REAL st S1[k0,d] )
assume A11: k0 in Seg m ; ::_thesis: ex d being Element of REAL st S1[k0,d]
reconsider k = k0 as Element of NAT by ORDINAL1:def_12;
A12: ( 1 <= k & k <= m ) by A11, FINSEQ_1:1;
then  f `partial| (X,k) is_continuous_on X by A1;
then consider d being Real such that
A13: ( 0 < d & ( for q being Element of REAL m st q in X & |.(q - x).| < d holds
|.(((f `partial| (X,k)) /. q) - ((f `partial| (X,k)) /. x)).| < (r0 / 2) / m ) ) by A9, A1, Th38;
take  d ; ::_thesis: S1[k0,d]
 for q being Element of REAL m st q in X & |.(q - x).| < d holds
|.((partdiff (f,q,k)) - (partdiff (f,x,k))).| < (r0 / 2) / m
proof
let q be Element of REAL m; ::_thesis: ( q in X & |.(q - x).| < d implies |.((partdiff (f,q,k)) - (partdiff (f,x,k))).| < (r0 / 2) / m )
assume A14: ( q in X & |.(q - x).| < d ) ; ::_thesis: |.((partdiff (f,q,k)) - (partdiff (f,x,k))).| < (r0 / 2) / m
then A15: |.(((f `partial| (X,k)) /. q) - ((f `partial| (X,k)) /. x)).| < (r0 / 2) / m by A13;
A16: f is_partial_differentiable_on X,k by A1, A12;
then  (f `partial| (X,k)) /. q = partdiff (f,q,k) by A14, Def5;
hence  |.((partdiff (f,q,k)) - (partdiff (f,x,k))).| < (r0 / 2) / m by A15, A16, A1, Def5; ::_thesis: verum
end;
hence  ex k being Element of NAT st
( k0 = k & 0 < d & ( for q being Element of REAL m st q in X & |.(q - x).| < d holds
|.((partdiff (f,q,k)) - (partdiff (f,x,k))).| < (r0 / 2) / m ) ) by A13; ::_thesis: verum
end;
consider Dseq being FinSequence of REAL  such that
A17: ( dom Dseq = Seg m & ( for i being Nat st i in Seg m holds
S1[i,Dseq . i] ) ) from FINSEQ_1:sch_5(A10);
A18: rng Dseq is finite by A17, FINSET_1:8;
 m in Seg m by FINSEQ_1:3;
then reconsider rDseq = rng Dseq as non empty ext-real-membered set by A17, FUNCT_1:3;
reconsider rDseq = rDseq as non empty ext-real-membered left_end right_end set by A18;
A19: min rDseq in rng Dseq by XXREAL_2:def_7;
then reconsider d1 = min rDseq as Real ;
set d = min (d0,d1);
consider i1 being set  such that
A20: ( i1 in dom Dseq & d1 = Dseq . i1 ) by A19, FUNCT_1:def_3;
reconsider i1 = i1 as Nat by A20;
A21: ex k being Element of NAT st
( i1 = k & 0 < Dseq . i1 & ( for q being Element of REAL m st q in X & |.(q - x).| < Dseq . i1 holds
|.((partdiff (f,q,k)) - (partdiff (f,x,k))).| < (r0 / 2) / m ) ) by A17, A20;
A22: now__::_thesis:_for_q_being_Element_of_REAL_m_st_|.(q_-_x).|_<_min_(d0,d1)_holds_
q_in_X
let q be Element of REAL m; ::_thesis: ( |.(q - x).| < min (d0,d1) implies q in X )
assume A23: |.(q - x).| < min (d0,d1) ; ::_thesis: q in X
 min (d0,d1) <= d0 by XXREAL_0:17;
then  |.(q - x).| < d0 by A23, XXREAL_0:2;
then  q in  {  y where y is Element of REAL m : |.(y - x).| < d0  }  ;
hence  q in X by A4; ::_thesis: verum
end;
A24: now__::_thesis:_for_q_being_Element_of_REAL_m
for_i_being_Element_of_NAT_st_|.(q_-_x).|_<_min_(d0,d1)_&_i_in_Seg_m_holds_
|.((partdiff_(f,q,i))_-_(partdiff_(f,x,i))).|_<_(r0_/_2)_/_m
let q be Element of REAL m; ::_thesis: for i being Element of NAT st |.(q - x).| < min (d0,d1) & i in Seg m holds
|.((partdiff (f,q,i)) - (partdiff (f,x,i))).| < (r0 / 2) / m
let i be Element of NAT ; ::_thesis: ( |.(q - x).| < min (d0,d1) & i in Seg m implies |.((partdiff (f,q,i)) - (partdiff (f,x,i))).| < (r0 / 2) / m )
assume A25: ( |.(q - x).| < min (d0,d1) & i in Seg m ) ; ::_thesis: |.((partdiff (f,q,i)) - (partdiff (f,x,i))).| < (r0 / 2) / m
reconsider i0 = i as Nat ;
consider k being Element of NAT  such that
A26: ( i0 = k & 0 < Dseq . i0 & ( for q being Element of REAL m st q in X & |.(q - x).| < Dseq . i0 holds
|.((partdiff (f,q,k)) - (partdiff (f,x,k))).| < (r0 / 2) / m ) ) by A17, A25;
 Dseq . i0 in rng Dseq by A17, A25, FUNCT_1:3;
then A27: d1 <= Dseq . i0 by XXREAL_2:def_7;
 min (d0,d1) <= d1 by XXREAL_0:17;
then  min (d0,d1) <= Dseq . i0 by A27, XXREAL_0:2;
then  |.(q - x).| < Dseq . i0 by A25, XXREAL_0:2;
hence  |.((partdiff (f,q,i)) - (partdiff (f,x,i))).| < (r0 / 2) / m by A22, A25, A26; ::_thesis: verum
end;
take d = min (d0,d1); ::_thesis: ( 0 < d & ( for y being Element of REAL m
for z being Element of REAL 1 st y <> 0* m & |.y.| < d & z = R . y holds
(|.y.| ") * |.z.| < r0 ) )
thus  0 < d by A3, A20, A21, XXREAL_0:21; ::_thesis: for y being Element of REAL m
for z being Element of REAL 1 st y <> 0* m & |.y.| < d & z = R . y holds
(|.y.| ") * |.z.| < r0
thus  for y being Element of REAL m
for z being Element of REAL 1 st y <> 0* m & |.y.| < d & z = R . y holds
(|.y.| ") * |.z.| < r0 ::_thesis: verum
proof
let y be Element of REAL m; ::_thesis: for z being Element of REAL 1 st y <> 0* m & |.y.| < d & z = R . y holds
(|.y.| ") * |.z.| < r0
let z be Element of REAL 1; ::_thesis: ( y <> 0* m & |.y.| < d & z = R . y implies (|.y.| ") * |.z.| < r0 )
assume A28: ( y <> 0* m & |.y.| < d & z = R . y ) ; ::_thesis: (|.y.| ") * |.z.| < r0
consider h being FinSequence of REAL m, g being FinSequence of REAL 1 such that
A29: ( len h = m + 1 & len g = m & ( for i being Nat st i in dom h holds
h /. i = (y | ((m + 1) -' i)) ^ (0* (i -' 1)) ) & ( for i being Nat st i in dom g holds
g /. i = (f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))) ) & ( for i being Nat
for hi being Element of REAL m st i in dom h & h /. i = hi holds
|.hi.| <= |.y.| ) & (f /. (x + y)) - (f /. x) = Sum g ) by Th28;
A30: R /. y = (Sum g) - (L . y) by A6, A29;
consider w being FinSequence of REAL 1 such that
A31: ( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = ((proj (i,m)) . y) * (partdiff (f,x,i)) ) & L . y = Sum w ) by A2;
 idseq m is Permutation of (Seg m) by FINSEQ_2:55;
then A32: ( dom (idseq m) = Seg m & rng (idseq m) = Seg m & idseq m is one-to-one ) by FUNCT_2:def_1, FUNCT_2:def_3;
then A33: ( dom (Rev (idseq m)) = Seg m & rng (Rev (idseq m)) = Seg m ) by FINSEQ_5:57;
then reconsider Ri = Rev (idseq m) as Function of (Seg m),(Seg m) by FUNCT_2:1;
 ( Ri is one-to-one & Ri is onto ) by A33, FUNCT_2:def_3;
then reconsider Ri = Rev (idseq m) as Permutation of (dom w) by A31;
A34: len (idseq m) = m by A32, FINSEQ_1:def_3
.= len w by A31, FINSEQ_1:def_3 ;
A35: dom (w (*) Ri) = dom Ri by A33, RELAT_1:27
.= dom (Rev w) by A33, A31, FINSEQ_5:57 ;
now__::_thesis:_for_k_being_Nat_st_k_in_dom_(Rev_w)_holds_
(Rev_w)_._k_=_(w_(*)_Ri)_._k
let k be Nat; ::_thesis: ( k in dom (Rev w) implies (Rev w) . k = (w (*) Ri) . k )
assume A36: k in dom (Rev w) ; ::_thesis: (Rev w) . k = (w (*) Ri) . k
then A37: k in dom (Rev (idseq m)) by A33, A31, FINSEQ_5:57;
then A38: ( 1 <= k & k <= m ) by A33, FINSEQ_1:1;
then reconsider mk = m - k as Nat by NAT_1:21;
A39: len (idseq m) = m by A32, FINSEQ_1:def_3;
 0 <= mk ;
then A40: 0 + 1 <= (m - k) + 1 by XREAL_1:6;
 k - 1 >= 1 - 1 by A38, XREAL_1:9;
then  m - (k - 1) <= m by XREAL_1:43;
then A41: mk + 1 in Seg m by A40;
thus (Rev w) . k = w . (((len (idseq m)) - k) + 1) by A34, A36, FINSEQ_5:def_3
.= w . ((idseq m) . (((len (idseq m)) - k) + 1)) by A41, A39, FINSEQ_2:49
.= w . ((Rev (idseq m)) . k) by A37, FINSEQ_5:def_3
.= (w (*) Ri) . k by A36, A35, FUNCT_1:12 ; ::_thesis: verum
end;
then A42: Sum (Rev w) = Sum w by A35, EUCLID_7:21, FINSEQ_1:13;
deffunc H2( Nat) ->  Element of REAL 1 = (g /. $1) - ((Rev w) /. $1);
consider gw being FinSequence of REAL 1 such that
A43: ( len gw = m & ( for j being Nat st j in dom gw holds
gw . j = H2(j) ) ) from FINSEQ_2:sch_1();
A44: now__::_thesis:_for_j_being_Nat_st_j_in_dom_gw_holds_
gw_/._j_=_(g_/._j)_-_((Rev_w)_/._j)
let j be Nat; ::_thesis: ( j in dom gw implies gw /. j = (g /. j) - ((Rev w) /. j) )
assume A45: j in dom gw ; ::_thesis: gw /. j = (g /. j) - ((Rev w) /. j)
hence gw /. j = gw . j by PARTFUN1:def_6
.= (g /. j) - ((Rev w) /. j) by A45, A43 ;
::_thesis: verum
end;
A46: len w = m by A31, FINSEQ_1:def_3;
then  len (Rev w) = m by FINSEQ_5:def_3;
then A47: R /. y = Sum gw by A29, A30, A31, A43, A44, A42, Th27;
A48: for j being Element of NAT st j in dom gw holds
ex gwj being Element of REAL 1 st
( gw . j = gwj & |.gwj.| <= |.y.| * ((r0 / 2) / m) )
proof
let j be Element of NAT ; ::_thesis: ( j in dom gw implies ex gwj being Element of REAL 1 st
( gw . j = gwj & |.gwj.| <= |.y.| * ((r0 / 2) / m) ) )
assume A49: j in dom gw ; ::_thesis: ex gwj being Element of REAL 1 st
( gw . j = gwj & |.gwj.| <= |.y.| * ((r0 / 2) / m) )
then A50: j in Seg m by A43, FINSEQ_1:def_3;
then  j in dom g by A29, FINSEQ_1:def_3;
then A51: g /. j = (f /. (x + (h /. j))) - (f /. (x + (h /. (j + 1)))) by A29;
A52: ( 1 <= j & j <= m ) by A50, FINSEQ_1:1;
then  ( m + 1 <= m + j & j + 1 <= m + 1 ) by XREAL_1:6;
then  ( (m + 1) - j <= m & 1 <= (m + 1) - j ) by XREAL_1:19, XREAL_1:20;
then A53: ( (m + 1) -' j <= m & 1 <= (m + 1) -' j ) by A52, NAT_D:37;
then A54: f is_partial_differentiable_on X,(m + 1) -' j by A1;
then A55: ( X c= dom f & ( for x being Element of REAL m st x in X holds
f is_partial_differentiable_in x,(m + 1) -' j ) ) by A53, Th34, A1;
A56: (m + 1) -' j in Seg m by A53;
then A57: w /. ((m + 1) -' j) = w . ((m + 1) -' j) by A31, PARTFUN1:def_6
.= ((proj (((m + 1) -' j),m)) . y) * (partdiff (f,x,((m + 1) -' j))) by A31, A56 ;
A58: now__::_thesis:_for_j_being_Element_of_NAT_st_1_<=_j_&_j_<=_m_+_1_holds_
|.((x_+_(h_/._j))_-_x).|_<_d
let j be Element of NAT ; ::_thesis: ( 1 <= j & j <= m + 1 implies |.((x + (h /. j)) - x).| < d )
assume  ( 1 <= j & j <= m + 1 ) ; ::_thesis: |.((x + (h /. j)) - x).| < d
then  j in Seg (m + 1) ;
then A59: j in dom h by A29, FINSEQ_1:def_3;
A60: (x + (h /. j)) - x = h /. j by RVSUM_1:42;
reconsider hj = h /. j as Element of REAL m ;
 |.hj.| <= |.y.| by A29, A59;
hence  |.((x + (h /. j)) - x).| < d by A60, A28, XXREAL_0:2; ::_thesis: verum
end;
 rng f0 c= dom ((proj (1,1)) ") by PDIFF_1:2;
then A61: dom f = dom f0 by A7, RELAT_1:27;
 m <= m + 1 by NAT_1:11;
then  Seg m c= Seg (m + 1) by FINSEQ_1:5;
then  ( 1 <= j & j <= m + 1 ) by A50, FINSEQ_1:1;
then A62: |.((x + (h /. j)) - x).| < d by A58;
then A63: x + (h /. j) in X by A22;
then A64: f /. (x + (h /. j)) = (<>* f0) . (x + (h /. j)) by A55, A7, PARTFUN1:def_6
.= ((proj (1,1)) ") . (f0 . (x + (h /. j))) by A63, A55, A61, FUNCT_1:13
.= <*(f0 . (x + (h /. j)))*> by PDIFF_1:1
.= <*(f0 /. (x + (h /. j)))*> by A63, A55, A61, PARTFUN1:def_6 ;
A65: ( 1 <= j & j <= m ) by A50, FINSEQ_1:1;
A66: 1 <= j + 1 by NAT_1:11;
A67: j + 1 <= m + 1 by A65, XREAL_1:6;
then A68: |.((x + (h /. (j + 1))) - x).| < d by A66, A58;
then A69: x + (h /. (j + 1)) in X by A22;
then A70: f /. (x + (h /. (j + 1))) = (<>* f0) . (x + (h /. (j + 1))) by A55, A7, PARTFUN1:def_6
.= ((proj (1,1)) ") . (f0 . (x + (h /. (j + 1)))) by A69, A55, A61, FUNCT_1:13
.= <*(f0 . (x + (h /. (j + 1))))*> by PDIFF_1:1
.= <*(f0 /. (x + (h /. (j + 1))))*> by A69, A55, A61, PARTFUN1:def_6 ;
 f is_partial_differentiable_in x,(m + 1) -' j by A54, A53, Th34, A1;
then A71: partdiff (f,x,((m + 1) -' j)) = <*(partdiff (f0,x,((m + 1) -' j)))*> by A7, PDIFF_1:19;
then A72: ((proj (((m + 1) -' j),m)) . y) * (partdiff (f,x,((m + 1) -' j))) = <*(((proj (((m + 1) -' j),m)) . y) * (partdiff (f0,x,((m + 1) -' j))))*> by RVSUM_1:47;
A73: (f /. (x + (h /. j))) - (f /. (x + (h /. (j + 1)))) = <*((f0 /. (x + (h /. j))) - (f0 /. (x + (h /. (j + 1)))))*> by A64, A70, RVSUM_1:29;
A74: ( X c= dom f0 & ( for x being Element of REAL m st x in X holds
f0 is_partial_differentiable_in x,(m + 1) -' j ) )
proof
thus  X c= dom f0 by A54, A53, Th34, A1, A61; ::_thesis: for x being Element of REAL m st x in X holds
f0 is_partial_differentiable_in x,(m + 1) -' j
let x be Element of REAL m; ::_thesis: ( x in X implies f0 is_partial_differentiable_in x,(m + 1) -' j )
assume  x in X ; ::_thesis: f0 is_partial_differentiable_in x,(m + 1) -' j
then  f is_partial_differentiable_in x,(m + 1) -' j by A54, A53, Th34, A1;
hence  f0 is_partial_differentiable_in x,(m + 1) -' j by A7, PDIFF_1:18; ::_thesis: verum
end;
A75: x + (h /. j) = (reproj (((m + 1) -' j),(x + (h /. (j + 1))))) . ((proj (((m + 1) -' j),m)) . (x + y)) by Th46, A29, A65;
 (m + 1) -' (j + 1) = (m + 1) - (j + 1) by A67, XREAL_1:233;
then  (m + 1) -' (j + 1) = m - j ;
then A76: (m + 1) -' (j + 1) = m -' j by A65, XREAL_1:233;
A77: (j + 1) -' 1 = (j + 1) - 1 by NAT_1:11, XREAL_1:233;
consider q being Element of REAL m such that
A78: ( |.(q - x).| < d & f0 is_partial_differentiable_in q,(m + 1) -' j & (f0 /. (x + (h /. j))) - (f0 /. (x + (h /. (j + 1)))) = (((proj (((m + 1) -' j),m)) . (x + y)) - ((proj (((m + 1) -' j),m)) . (x + (h /. (j + 1))))) * (partdiff (f0,q,((m + 1) -' j))) ) by A62, A68, A75, A53, A74, A22, A1, Th45;
A79: |.((partdiff (f,q,((m + 1) -' j))) - (partdiff (f,x,((m + 1) -' j)))).| < (r0 / 2) / m by A78, A56, A24;
 f is_partial_differentiable_in q,(m + 1) -' j by A78, A7, PDIFF_1:18;
then A80: partdiff (f,q,((m + 1) -' j)) = <*(partdiff (f0,q,((m + 1) -' j)))*> by A7, PDIFF_1:19;
set mij = (m + 1) -' j;
set mj = m -' j;
reconsider hj1 = h /. (j + 1) as Element of REAL m ;
A81: ( len x = m & len y = m & len hj1 = m ) by CARD_1:def_7;
then  ( (m + 1) -' j in dom x & (m + 1) -' j in dom y & (m + 1) -' j in dom hj1 ) by A56, FINSEQ_1:def_3;
then  ( (m + 1) -' j in (dom x) /\ (dom y) & (m + 1) -' j in (dom x) /\ (dom hj1) ) by XBOOLE_0:def_4;
then A82: ( (m + 1) -' j in dom (x + y) & (m + 1) -' j in dom (x + hj1) ) by VALUED_1:def_1;
 j + 1 in Seg (m + 1) by A66, A67;
then  j + 1 in dom h by A29, FINSEQ_1:def_3;
then A83: h /. (j + 1) = (y | (m -' j)) ^ (0* j) by A29, A76, A77;
A84: len (y | (m -' j)) = m -' j by A81, FINSEQ_1:59, NAT_D:35;
 (m + 1) -' j = (m -' j) + 1 by A65, NAT_D:38;
then  (m + 1) -' j > len (y | (m -' j)) by A84, NAT_1:13;
then A85: hj1 . ((m + 1) -' j) = (0* j) . (((m + 1) -' j) - (m -' j)) by A53, A81, A83, A84, FINSEQ_1:24
.= 0 ;
A86: ((proj (((m + 1) -' j),m)) . (x + y)) - ((proj (((m + 1) -' j),m)) . (x + (h /. (j + 1)))) = ((x + y) . ((m + 1) -' j)) - ((proj (((m + 1) -' j),m)) . (x + (h /. (j + 1)))) by PDIFF_1:def_1
.= ((x + y) . ((m + 1) -' j)) - ((x + (h /. (j + 1))) . ((m + 1) -' j)) by PDIFF_1:def_1
.= ((x . ((m + 1) -' j)) + (y . ((m + 1) -' j))) - ((x + (h /. (j + 1))) . ((m + 1) -' j)) by A82, VALUED_1:def_1
.= ((x . ((m + 1) -' j)) + (y . ((m + 1) -' j))) - ((x . ((m + 1) -' j)) + 0) by A85, A82, VALUED_1:def_1
.= (proj (((m + 1) -' j),m)) . y by PDIFF_1:def_1 ;
reconsider gwj = gw /. j as Element of REAL 1 ;
take  gwj ; ::_thesis: ( gw . j = gwj & |.gwj.| <= |.y.| * ((r0 / 2) / m) )
thus  gw . j = gwj by A49, PARTFUN1:def_6; ::_thesis: |.gwj.| <= |.y.| * ((r0 / 2) / m)
A87: (m + 1) -' j = (m + 1) - j by A65, NAT_1:12, XREAL_1:233;
 j in Seg (len (Rev w)) by A50, A46, FINSEQ_5:def_3;
then A88: j in dom (Rev w) by FINSEQ_1:def_3;
then (Rev w) /. j = (Rev w) . j by PARTFUN1:def_6
.= w . ((m - j) + 1) by A46, A88, FINSEQ_5:def_3
.= w /. ((m + 1) -' j) by A87, A56, A31, PARTFUN1:def_6 ;
then gw /. j = (g /. j) - (w /. ((m + 1) -' j)) by A49, A44
.= <*((((proj (((m + 1) -' j),m)) . y) * (partdiff (f0,q,((m + 1) -' j)))) - (((proj (((m + 1) -' j),m)) . y) * (partdiff (f0,x,((m + 1) -' j)))))*> by A78, A86, A57, A51, A72, A73, RVSUM_1:29
.= <*(((proj (((m + 1) -' j),m)) . y) * ((partdiff (f0,q,((m + 1) -' j))) - (partdiff (f0,x,((m + 1) -' j)))))*>
.= ((proj (((m + 1) -' j),m)) . y) * <*((partdiff (f0,q,((m + 1) -' j))) - (partdiff (f0,x,((m + 1) -' j))))*> by RVSUM_1:47
.= ((proj (((m + 1) -' j),m)) . y) * ((partdiff (f,q,((m + 1) -' j))) - (partdiff (f,x,((m + 1) -' j)))) by A71, A80, RVSUM_1:29 ;
then A89: |.gwj.| = (abs ((proj (((m + 1) -' j),m)) . y)) * |.((partdiff (f,q,((m + 1) -' j))) - (partdiff (f,x,((m + 1) -' j)))).| by EUCLID:11;
 0 <= abs ((proj (((m + 1) -' j),m)) . y) by COMPLEX1:46;
then A90: |.gwj.| <= (abs ((proj (((m + 1) -' j),m)) . y)) * ((r0 / 2) / m) by A89, A79, XREAL_1:64;
 abs (y . ((m + 1) -' j)) <= |.y.| by A56, REAL_NS1:8;
then  abs ((proj (((m + 1) -' j),m)) . y) <= |.y.| by PDIFF_1:def_1;
then  (abs ((proj (((m + 1) -' j),m)) . y)) * ((r0 / 2) / m) <= |.y.| * ((r0 / 2) / m) by A9, XREAL_1:64;
hence  |.gwj.| <= |.y.| * ((r0 / 2) / m) by A90, XXREAL_0:2; ::_thesis: verum
end;
defpred S2[ set , set ] means  ex v being Element of REAL 1 st
( v = gw . $1 & $2 = |.v.| );
A91: now__::_thesis:_for_k_being_Nat_st_k_in_Seg_m_holds_
ex_x_being_Element_of_REAL_st_S2[k,x]
let k be Nat; ::_thesis: ( k in Seg m implies ex x being Element of REAL st S2[k,x] )
assume  k in Seg m ; ::_thesis: ex x being Element of REAL st S2[k,x]
then  k in dom gw by A43, FINSEQ_1:def_3;
then reconsider v = gw . k as Element of REAL 1 by PARTFUN1:4;
 |.v.| in REAL ;
hence  ex x being Element of REAL st S2[k,x] ; ::_thesis: verum
end;
consider yseq being FinSequence of REAL  such that
A92: ( dom yseq = Seg m & ( for i being Nat st i in Seg m holds
S2[i,yseq . i] ) ) from FINSEQ_1:sch_5(A91);
A93: len gw = len yseq by A43, A92, FINSEQ_1:def_3;
A94: now__::_thesis:_for_i_being_Element_of_NAT_st_i_in_dom_gw_holds_
ex_v_being_Element_of_REAL_1_st_
(_v_=_gw_._i_&_yseq_._i_=_|.v.|_)
let i be Element of NAT ; ::_thesis: ( i in dom gw implies ex v being Element of REAL 1 st
( v = gw . i & yseq . i = |.v.| ) )
assume  i in dom gw ; ::_thesis: ex v being Element of REAL 1 st
( v = gw . i & yseq . i = |.v.| )
then  i in Seg m by A43, FINSEQ_1:def_3;
hence  ex v being Element of REAL 1 st
( v = gw . i & yseq . i = |.v.| ) by A92; ::_thesis: verum
end;
reconsider yseq = yseq as Element of REAL m by A93, A43, FINSEQ_2:92;
A95: |.(Sum gw).| <= Sum yseq by A94, A93, PDIFF_6:17;
 for j being Nat st j in Seg m holds
yseq . j <= (m |-> (|.y.| * ((r0 / 2) / m))) . j
proof
let j be Nat; ::_thesis: ( j in Seg m implies yseq . j <= (m |-> (|.y.| * ((r0 / 2) / m))) . j )
assume A96: j in Seg m ; ::_thesis: yseq . j <= (m |-> (|.y.| * ((r0 / 2) / m))) . j
then A97: j in dom gw by A43, FINSEQ_1:def_3;
then A98: ex v being Element of REAL 1 st
( v = gw . j & yseq . j = |.v.| ) by A94;
 ex gwj being Element of REAL 1 st
( gw . j = gwj & |.gwj.| <= |.y.| * ((r0 / 2) / m) ) by A48, A97;
hence  yseq . j <= (m |-> (|.y.| * ((r0 / 2) / m))) . j by A98, A96, FINSEQ_2:57; ::_thesis: verum
end;
then  Sum yseq <= Sum (m |-> (|.y.| * ((r0 / 2) / m))) by RVSUM_1:82;
then  Sum yseq <= m * (|.y.| * ((r0 / 2) / m)) by RVSUM_1:80;
then  |.z.| <= m * (|.y.| * ((r0 / 2) / m)) by A47, A28, A95, XXREAL_0:2;
then  |.z.| * (|.y.| ") <= ((m * |.y.|) * ((r0 / 2) / m)) * (|.y.| ") by XREAL_1:64;
then  |.z.| * (|.y.| ") <= m * ((((r0 / 2) / m) * |.y.|) * (|.y.| ")) ;
then  (|.y.| ") * |.z.| <= m * ((r0 / 2) / m) by A28, EUCLID:8, XCMPLX_1:203;
then A99: (|.y.| ") * |.z.| <= r0 / 2 by XCMPLX_1:87;
 r0 / 2 < r0 by A9, XREAL_1:216;
hence  (|.y.| ") * |.z.| < r0 by A99, XXREAL_0:2; ::_thesis: verum
end;
end;
 for y being Element of REAL m st |.(y - x).| < d0 holds
(f /. y) - (f /. x) = (L . (y - x)) + (R . (y - x))
proof
let y be Element of REAL m; ::_thesis: ( |.(y - x).| < d0 implies (f /. y) - (f /. x) = (L . (y - x)) + (R . (y - x)) )
assume  |.(y - x).| < d0 ; ::_thesis: (f /. y) - (f /. x) = (L . (y - x)) + (R . (y - x))
 R . (y - x) = ((f /. (x + (y - x))) - (f /. x)) - (L . (y - x)) by A6;
hence (L . (y - x)) + (R . (y - x)) = ((f /. (x + (y - x))) - (f /. x)) - ((L . (y - x)) - (L . (y - x))) by RVSUM_1:41
.= ((f /. (x + (y - x))) - (f /. x)) - (0* 1) by RVSUM_1:37
.= (f /. (x + (y - x))) - (f /. x) by RVSUM_1:32
.= (f /. ((x + y) - x)) - (f /. x) by RVSUM_1:40
.= (f /. (y + (x - x))) - (f /. x) by RVSUM_1:40
.= (f /. (y + (0* m))) - (f /. x) by RVSUM_1:37
.= (f /. y) - (f /. x) by RVSUM_1:16 ;
::_thesis: verum
end;
then  ( f is_differentiable_in x & diff (f,x) = L ) by A3, A5, A8, PDIFF_6:24;
hence  ( f is_differentiable_in x & ( for h being Element of REAL m ex w being FinSequence of REAL 1 st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = ((proj (i,m)) . h) * (partdiff (f,x,i)) ) & (diff (f,x)) . h = Sum w ) ) ) by A2; ::_thesis: verum
end;

theorem Th48: :: PDIFF_7:48
for m being non empty Element of NAT
for f being PartFunc of (REAL-NS m),(REAL-NS 1)
for X being Subset of (REAL-NS m)
for x being Point of (REAL-NS m) st X is open & x in X & ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) holds
( f is_differentiable_in x & ( for h being Point of (REAL-NS m) ex w being FinSequence of REAL 1 st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = (partdiff (f,x,i)) . <*((proj (i,m)) . h)*> ) & (diff (f,x)) . h = Sum w ) ) )
proof
let m be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL-NS m),(REAL-NS 1)
for X being Subset of (REAL-NS m)
for x being Point of (REAL-NS m) st X is open & x in X & ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) holds
( f is_differentiable_in x & ( for h being Point of (REAL-NS m) ex w being FinSequence of REAL 1 st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = (partdiff (f,x,i)) . <*((proj (i,m)) . h)*> ) & (diff (f,x)) . h = Sum w ) ) )
let f be PartFunc of (REAL-NS m),(REAL-NS 1); ::_thesis: for X being Subset of (REAL-NS m)
for x being Point of (REAL-NS m) st X is open & x in X & ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) holds
( f is_differentiable_in x & ( for h being Point of (REAL-NS m) ex w being FinSequence of REAL 1 st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = (partdiff (f,x,i)) . <*((proj (i,m)) . h)*> ) & (diff (f,x)) . h = Sum w ) ) )
let X be Subset of (REAL-NS m); ::_thesis: for x being Point of (REAL-NS m) st X is open & x in X & ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) holds
( f is_differentiable_in x & ( for h being Point of (REAL-NS m) ex w being FinSequence of REAL 1 st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = (partdiff (f,x,i)) . <*((proj (i,m)) . h)*> ) & (diff (f,x)) . h = Sum w ) ) )
let x be Point of (REAL-NS m); ::_thesis: ( X is open & x in X & ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) implies ( f is_differentiable_in x & ( for h being Point of (REAL-NS m) ex w being FinSequence of REAL 1 st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = (partdiff (f,x,i)) . <*((proj (i,m)) . h)*> ) & (diff (f,x)) . h = Sum w ) ) ) )
assume A1: ( X is open & x in X & ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) ) ; ::_thesis: ( f is_differentiable_in x & ( for h being Point of (REAL-NS m) ex w being FinSequence of REAL 1 st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = (partdiff (f,x,i)) . <*((proj (i,m)) . h)*> ) & (diff (f,x)) . h = Sum w ) ) )
A2: ( the carrier of (REAL-NS m) = REAL m & the carrier of (REAL-NS 1) = REAL 1 ) by REAL_NS1:def_4;
reconsider One0 = <*1*> as Element of REAL 1 by FINSEQ_2:98;
reconsider One = One0 as Point of (REAL-NS 1) by REAL_NS1:def_4;
reconsider f0 = f as PartFunc of (REAL m),(REAL 1) by A2;
reconsider X0 = X as Subset of (REAL m) by REAL_NS1:def_4;
reconsider x0 = x as Element of REAL m by REAL_NS1:def_4;
A3: X0 is open by Def3, A1;
A4: now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_m_holds_
(_f0_is_partial_differentiable_on_X0,i_&_f0_`partial|_(X0,i)_is_continuous_on_X0_)
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m implies ( f0 is_partial_differentiable_on X0,i & f0 `partial| (X0,i) is_continuous_on X0 ) )
assume A5: ( 1 <= i & i <= m ) ; ::_thesis: ( f0 is_partial_differentiable_on X0,i & f0 `partial| (X0,i) is_continuous_on X0 )
then A6: f is_partial_differentiable_on X,i by A1;
hence A7: f0 is_partial_differentiable_on X0,i by Th33; ::_thesis: f0 `partial| (X0,i) is_continuous_on X0
A8: f `partial| (X,i) is_continuous_on X by A1, A5;
A9: ( dom (f0 `partial| (X0,i)) = X0 & ( for x0 being Element of REAL m st x0 in X0 holds
(f0 `partial| (X0,i)) /. x0 = partdiff (f0,x0,i) ) ) by Def5, A7;
A10: for z being Element of REAL m st z in X0 holds
ex y being Point of (REAL-NS m) st
( z = y & (f0 `partial| (X0,i)) /. z = (partdiff (f,y,i)) . One )
proof
let z be Element of REAL m; ::_thesis: ( z in X0 implies ex y being Point of (REAL-NS m) st
( z = y & (f0 `partial| (X0,i)) /. z = (partdiff (f,y,i)) . One ) )
assume A11: z in X0 ; ::_thesis: ex y being Point of (REAL-NS m) st
( z = y & (f0 `partial| (X0,i)) /. z = (partdiff (f,y,i)) . One )
then  f0 is_partial_differentiable_in z,i by A7, A5, A3, Th34;
then consider g being PartFunc of (REAL-NS m),(REAL-NS 1), y being Point of (REAL-NS m) such that
A12: ( f0 = g & z = y & partdiff (f0,z,i) = (partdiff (g,y,i)) . <*1*> ) by PDIFF_1:def_14;
take  y ; ::_thesis: ( z = y & (f0 `partial| (X0,i)) /. z = (partdiff (f,y,i)) . One )
thus  z = y by A12; ::_thesis: (f0 `partial| (X0,i)) /. z = (partdiff (f,y,i)) . One
thus  (f0 `partial| (X0,i)) /. z = (partdiff (f,y,i)) . One by A12, A11, Def5, A7; ::_thesis: verum
end;
 for z0 being Element of REAL m
for r being Real st z0 in X0 & 0 < r holds
ex s being Real st
( 0 < s & ( for z1 being Element of REAL m st z1 in X & |.(z1 - z0).| < s holds
|.(((f0 `partial| (X0,i)) /. z1) - ((f0 `partial| (X0,i)) /. z0)).| < r ) )
proof
let z0 be Element of REAL m; ::_thesis: for r being Real st z0 in X0 & 0 < r holds
ex s being Real st
( 0 < s & ( for z1 being Element of REAL m st z1 in X & |.(z1 - z0).| < s holds
|.(((f0 `partial| (X0,i)) /. z1) - ((f0 `partial| (X0,i)) /. z0)).| < r ) )
let r be Real; ::_thesis: ( z0 in X0 & 0 < r implies ex s being Real st
( 0 < s & ( for z1 being Element of REAL m st z1 in X & |.(z1 - z0).| < s holds
|.(((f0 `partial| (X0,i)) /. z1) - ((f0 `partial| (X0,i)) /. z0)).| < r ) ) )
assume A13: ( z0 in X0 & 0 < r ) ; ::_thesis: ex s being Real st
( 0 < s & ( for z1 being Element of REAL m st z1 in X & |.(z1 - z0).| < s holds
|.(((f0 `partial| (X0,i)) /. z1) - ((f0 `partial| (X0,i)) /. z0)).| < r ) )
reconsider y0 = z0 as Point of (REAL-NS m) by REAL_NS1:def_4;
consider s being Real such that
A14: ( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r ) ) by A8, A13, NFCONT_1:19;
take  s ; ::_thesis: ( 0 < s & ( for z1 being Element of REAL m st z1 in X & |.(z1 - z0).| < s holds
|.(((f0 `partial| (X0,i)) /. z1) - ((f0 `partial| (X0,i)) /. z0)).| < r ) )
thus  0 < s by A14; ::_thesis: for z1 being Element of REAL m st z1 in X & |.(z1 - z0).| < s holds
|.(((f0 `partial| (X0,i)) /. z1) - ((f0 `partial| (X0,i)) /. z0)).| < r
thus  for z1 being Element of REAL m st z1 in X & |.(z1 - z0).| < s holds
|.(((f0 `partial| (X0,i)) /. z1) - ((f0 `partial| (X0,i)) /. z0)).| < r ::_thesis: verum
proof
let z1 be Element of REAL m; ::_thesis: ( z1 in X & |.(z1 - z0).| < s implies |.(((f0 `partial| (X0,i)) /. z1) - ((f0 `partial| (X0,i)) /. z0)).| < r )
assume A15: ( z1 in X & |.(z1 - z0).| < s ) ; ::_thesis: |.(((f0 `partial| (X0,i)) /. z1) - ((f0 `partial| (X0,i)) /. z0)).| < r
reconsider y1 = z1 as Point of (REAL-NS m) by REAL_NS1:def_4;
 |.(z1 - z0).| = ||.(y1 - y0).|| by REAL_NS1:1, REAL_NS1:5;
then A16: ||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r by A15, A14;
 (f `partial| (X,i)) /. y1 = partdiff (f,y1,i) by A6, A15, PDIFF_1:def_20;
then A17: ||.((partdiff (f,y1,i)) - (partdiff (f,y0,i))).|| < r by A16, A6, A13, PDIFF_1:def_20;
A18: ex y1 being Point of (REAL-NS m) st
( z1 = y1 & (f0 `partial| (X0,i)) /. z1 = (partdiff (f,y1,i)) . One ) by A10, A15;
A19: ex y0 being Point of (REAL-NS m) st
( z0 = y0 & (f0 `partial| (X0,i)) /. z0 = (partdiff (f,y0,i)) . One ) by A10, A13;
reconsider PDP = (partdiff (f,y1,i)) - (partdiff (f,y0,i)) as Lipschitzian LinearOperator of (REAL-NS 1),(REAL-NS 1) by LOPBAN_1:def_9;
 ((partdiff (f,y1,i)) . One) - ((partdiff (f,y0,i)) . One) = PDP . One by LOPBAN_1:40;
then A20: ||.(((partdiff (f,y1,i)) . One) - ((partdiff (f,y0,i)) . One)).|| <= ||.((partdiff (f,y1,i)) - (partdiff (f,y0,i))).|| * ||.One.|| by LOPBAN_1:32;
||.One.|| = |.One0.| by REAL_NS1:1
.= abs 1 by TOPREALC:18
.= 1 by ABSVALUE:def_1 ;
then  ||.(((partdiff (f,y1,i)) . One) - ((partdiff (f,y0,i)) . One)).|| < r by A20, A17, XXREAL_0:2;
hence  |.(((f0 `partial| (X0,i)) /. z1) - ((f0 `partial| (X0,i)) /. z0)).| < r by A18, A19, REAL_NS1:1, REAL_NS1:5; ::_thesis: verum
end;
end;
hence  f0 `partial| (X0,i) is_continuous_on X0 by A9, Th38; ::_thesis: verum
end;
then A21: ( f0 is_differentiable_in x0 & ( for h0 being Element of REAL m ex w being FinSequence of REAL 1 st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = ((proj (i,m)) . h0) * (partdiff (f0,x0,i)) ) & (diff (f0,x0)) . h0 = Sum w ) ) ) by Th47, A3, A1;
then  ex g being PartFunc of (REAL-NS m),(REAL-NS 1) ex y being Point of (REAL-NS m) st
( f0 = g & x0 = y & g is_differentiable_in y ) by PDIFF_1:def_7;
hence  f is_differentiable_in x ; ::_thesis: for h being Point of (REAL-NS m) ex w being FinSequence of REAL 1 st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = (partdiff (f,x,i)) . <*((proj (i,m)) . h)*> ) & (diff (f,x)) . h = Sum w )
A22: ex g being PartFunc of (REAL-NS m),(REAL-NS 1) ex y being Point of (REAL-NS m) st
( f0 = g & x0 = y & diff (f0,x0) = diff (g,y) ) by A21, PDIFF_1:def_8;
now__::_thesis:_for_h_being_Point_of_(REAL-NS_m)_ex_w_being_FinSequence_of_REAL_1_st_
(_dom_w_=_Seg_m_&_(_for_i_being_Element_of_NAT_st_i_in_Seg_m_holds_
w_._i_=_(partdiff_(f,x,i))_._<*((proj_(i,m))_._h)*>_)_&_(diff_(f,x))_._h_=_Sum_w_)
let h be Point of (REAL-NS m); ::_thesis: ex w being FinSequence of REAL 1 st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = (partdiff (f,x,i)) . <*((proj (i,m)) . h)*> ) & (diff (f,x)) . h = Sum w )
reconsider h0 = h as Element of REAL m by REAL_NS1:def_4;
consider w being FinSequence of REAL 1 such that
A23: ( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = ((proj (i,m)) . h0) * (partdiff (f0,x0,i)) ) & (diff (f0,x0)) . h0 = Sum w ) by Th47, A3, A1, A4;
take w = w; ::_thesis: ( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = (partdiff (f,x,i)) . <*((proj (i,m)) . h)*> ) & (diff (f,x)) . h = Sum w )
thus  dom w = Seg m by A23; ::_thesis: ( ( for i being Element of NAT st i in Seg m holds
w . i = (partdiff (f,x,i)) . <*((proj (i,m)) . h)*> ) & (diff (f,x)) . h = Sum w )
thus  for i being Element of NAT st i in Seg m holds
w . i = (partdiff (f,x,i)) . <*((proj (i,m)) . h)*> ::_thesis: (diff (f,x)) . h = Sum w
proof
let i be Element of NAT ; ::_thesis: ( i in Seg m implies w . i = (partdiff (f,x,i)) . <*((proj (i,m)) . h)*> )
assume A24: i in Seg m ; ::_thesis: w . i = (partdiff (f,x,i)) . <*((proj (i,m)) . h)*>
then A25: ( 1 <= i & i <= m ) by FINSEQ_1:1;
then  f0 is_partial_differentiable_on X0,i by A4;
then  f0 is_partial_differentiable_in x0,i by A1, A3, A25, Th34;
then A26: ex g being PartFunc of (REAL-NS m),(REAL-NS 1) ex y being Point of (REAL-NS m) st
( f0 = g & x0 = y & partdiff (f0,x0,i) = (partdiff (g,y,i)) . <*1*> ) by PDIFF_1:def_14;
A27: ((proj (i,m)) . h) * One = ((proj (i,m)) . h0) * One0 by REAL_NS1:3
.= <*(((proj (i,m)) . h0) * 1)*> by RVSUM_1:47
.= <*((proj (i,m)) . h)*> ;
reconsider PDP = partdiff (f,x,i) as Lipschitzian LinearOperator of (REAL-NS 1),(REAL-NS 1) by LOPBAN_1:def_9;
((proj (i,m)) . h0) * (partdiff (f0,x0,i)) = ((proj (i,m)) . h0) * (PDP . One) by A26, REAL_NS1:3
.= (partdiff (f,x,i)) . <*((proj (i,m)) . h)*> by A27, LOPBAN_1:def_5 ;
hence  w . i = (partdiff (f,x,i)) . <*((proj (i,m)) . h)*> by A24, A23; ::_thesis: verum
end;
thus  (diff (f,x)) . h = Sum w by A23, A22; ::_thesis: verum
end;
hence  for h being Point of (REAL-NS m) ex w being FinSequence of REAL 1 st
( dom w = Seg m & ( for i being Element of NAT st i in Seg m holds
w . i = (partdiff (f,x,i)) . <*((proj (i,m)) . h)*> ) & (diff (f,x)) . h = Sum w ) ; ::_thesis: verum
end;

theorem :: PDIFF_7:49
for m being non empty Element of NAT
for f being PartFunc of (REAL-NS m),(REAL-NS 1)
for X being Subset of (REAL-NS m) st X is open holds
( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & f `| X is_continuous_on X ) )
proof
let m be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL-NS m),(REAL-NS 1)
for X being Subset of (REAL-NS m) st X is open holds
( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & f `| X is_continuous_on X ) )
let f be PartFunc of (REAL-NS m),(REAL-NS 1); ::_thesis: for X being Subset of (REAL-NS m) st X is open holds
( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & f `| X is_continuous_on X ) )
let X be Subset of (REAL-NS m); ::_thesis: ( X is open implies ( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & f `| X is_continuous_on X ) ) )
assume A1: X is open ; ::_thesis: ( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & f `| X is_continuous_on X ) )
hereby  ::_thesis: ( f is_differentiable_on X & f `| X is_continuous_on X implies for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) )
assume A2: for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ; ::_thesis: ( f is_differentiable_on X & f `| X is_continuous_on X )
A3: now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_m_holds_
(_X_c=_dom_(f_`partial|_(X,i))_&_(_for_y0_being_Point_of_(REAL-NS_m)
for_r_being_Real_st_y0_in_X_&_0_<_r_holds_
ex_s_being_Real_st_
(_0_<_s_&_(_for_y1_being_Point_of_(REAL-NS_m)_st_y1_in_X_&_||.(y1_-_y0).||_<_s_holds_
||.(((f_`partial|_(X,i))_/._y1)_-_((f_`partial|_(X,i))_/._y0)).||_<_r_)_)_)_)
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m implies ( X c= dom (f `partial| (X,i)) & ( for y0 being Point of (REAL-NS m)
for r being Real st y0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r ) ) ) ) )
assume  ( 1 <= i & i <= m ) ; ::_thesis: ( X c= dom (f `partial| (X,i)) & ( for y0 being Point of (REAL-NS m)
for r being Real st y0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r ) ) ) )
then  f `partial| (X,i) is_continuous_on X by A2;
hence  ( X c= dom (f `partial| (X,i)) & ( for y0 being Point of (REAL-NS m)
for r being Real st y0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r ) ) ) ) by NFCONT_1:19; ::_thesis: verum
end;
 1 <= m by NAT_1:14;
then  f is_partial_differentiable_on X,m by A2;
then A4: X c= dom f by PDIFF_1:def_19;
 for x being Point of (REAL-NS m) st x in X holds
f is_differentiable_in x by A1, A2, Th48;
hence A5: f is_differentiable_on X by A1, A4, NDIFF_1:31; ::_thesis: f `| X is_continuous_on X
then A6: dom (f `| X) = X by NDIFF_1:def_9;
 for y0 being Point of (REAL-NS m)
for r being Real st y0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r ) )
proof
let y0 be Point of (REAL-NS m); ::_thesis: for r being Real st y0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r ) )
let r be Real; ::_thesis: ( y0 in X & 0 < r implies ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r ) ) )
assume A7: ( y0 in X & 0 < r ) ; ::_thesis: ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r ) )
defpred S1[ Nat, Real] means  for i being Element of NAT st i = $1 holds
( 0 < $2 & ( for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < $2 holds
||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r / (2 * m) ) );
A8: now__::_thesis:_for_i_being_Nat_st_i_in_Seg_m_holds_
ex_s_being_Element_of_REAL_st_S1[i,s]
let i be Nat; ::_thesis: ( i in Seg m implies ex s being Element of REAL st S1[i,s] )
reconsider j = i as Element of NAT by ORDINAL1:def_12;
assume  i in Seg m ; ::_thesis: ex s being Element of REAL st S1[i,s]
then  ( 1 <= j & j <= m ) by FINSEQ_1:1;
then consider s being Real such that
A9: ( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `partial| (X,j)) /. y1) - ((f `partial| (X,j)) /. y0)).|| < r / (2 * m) ) ) by A7, A3;
reconsider s = s as Element of REAL ;
take s = s; ::_thesis: S1[i,s]
thus  S1[i,s] by A9; ::_thesis: verum
end;
consider S being FinSequence of REAL  such that
A10: ( dom S = Seg m & ( for i being Nat st i in Seg m holds
S1[i,S . i] ) ) from FINSEQ_1:sch_5(A8);
take s = min S; ::_thesis: ( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r ) )
A11: len S = m by A10, FINSEQ_1:def_3;
then  ( s = S . (min_p S) & min_p S in dom S ) by RFINSEQ2:def_2, RFINSEQ2:def_4;
hence  s > 0 by A10; ::_thesis: for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r
let y1 be Point of (REAL-NS m); ::_thesis: ( y1 in X & ||.(y1 - y0).|| < s implies ||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r )
assume A12: ( y1 in X & ||.(y1 - y0).|| < s ) ; ::_thesis: ||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r
reconsider DD = (diff (f,y1)) - (diff (f,y0)) as Lipschitzian LinearOperator of (REAL-NS m),(REAL-NS 1) by LOPBAN_1:def_9;
A13: upper_bound (PreNorms DD) = ||.((diff (f,y1)) - (diff (f,y0))).|| by LOPBAN_1:30;
now__::_thesis:_for_mt_being_real_number_st_mt_in_PreNorms_DD_holds_
mt_<=_r_/_2
let mt be real number ; ::_thesis: ( mt in PreNorms DD implies mt <= r / 2 )
assume  mt in PreNorms DD ; ::_thesis: mt <= r / 2
then consider t being VECTOR of (REAL-NS m) such that
A14: ( mt = ||.(DD . t).|| & ||.t.|| <= 1 ) ;
reconsider tt = t as Element of REAL m by REAL_NS1:def_4;
consider w0 being FinSequence of REAL 1 such that
A15: ( dom w0 = Seg m & ( for i being Element of NAT st i in Seg m holds
w0 . i = (partdiff (f,y0,i)) . <*((proj (i,m)) . t)*> ) & (diff (f,y0)) . t = Sum w0 ) by A1, A2, Th48, A7;
reconsider Sw0 = Sum w0 as Point of (REAL-NS 1) by A15;
consider w1 being FinSequence of REAL 1 such that
A16: ( dom w1 = Seg m & ( for i being Element of NAT st i in Seg m holds
w1 . i = (partdiff (f,y1,i)) . <*((proj (i,m)) . t)*> ) & (diff (f,y1)) . t = Sum w1 ) by A1, A2, Th48, A12;
reconsider Sw1 = Sum w1 as Point of (REAL-NS 1) by A16;
deffunc H1( set ) ->  Element of REAL 1 = (w1 /. $1) - (w0 /. $1);
consider w2 being FinSequence of REAL 1 such that
A17: ( len w2 = m & ( for i being Nat st i in dom w2 holds
w2 . i = H1(i) ) ) from FINSEQ_2:sch_1();
A18: ( len w1 = m & len w0 = m ) by A15, A16, FINSEQ_1:def_3;
now__::_thesis:_for_i_being_Nat_st_i_in_dom_w2_holds_
w2_/._i_=_H1(i)
let i be Nat; ::_thesis: ( i in dom w2 implies w2 /. i = H1(i) )
assume A19: i in dom w2 ; ::_thesis: w2 /. i = H1(i)
then  w2 . i = H1(i) by A17;
hence  w2 /. i = H1(i) by A19, PARTFUN1:def_6; ::_thesis: verum
end;
then A20: Sum w2 = (Sum w1) - (Sum w0) by A17, Th27, A18;
DD . t = Sw1 - Sw0 by A16, A15, LOPBAN_1:40
.= Sum w2 by A20, REAL_NS1:5 ;
then A21: mt = |.(Sum w2).| by A14, REAL_NS1:1;
deffunc H2( Nat) ->  Element of REAL = |.(w2 /. $1).|;
consider ys being FinSequence of REAL  such that
A22: ( len ys = m & ( for j being Nat st j in dom ys holds
ys . j = H2(j) ) ) from FINSEQ_2:sch_1();
A23: now__::_thesis:_for_i_being_Element_of_NAT_st_i_in_dom_w2_holds_
ex_v_being_Element_of_REAL_1_st_
(_v_=_w2_._i_&_ys_._i_=_|.v.|_)
let i be Element of NAT ; ::_thesis: ( i in dom w2 implies ex v being Element of REAL 1 st
( v = w2 . i & ys . i = |.v.| ) )
assume A24: i in dom w2 ; ::_thesis: ex v being Element of REAL 1 st
( v = w2 . i & ys . i = |.v.| )
reconsider v = w2 /. i as Element of REAL 1 ;
take v = v; ::_thesis: ( v = w2 . i & ys . i = |.v.| )
thus  v = w2 . i by A24, PARTFUN1:def_6; ::_thesis: ys . i = |.v.|
 i in Seg m by A17, A24, FINSEQ_1:def_3;
then  i in dom ys by A22, FINSEQ_1:def_3;
hence  ys . i = |.v.| by A22; ::_thesis: verum
end;
then A25: |.(Sum w2).| <= Sum ys by A17, A22, PDIFF_6:17;
reconsider rm = r / (2 * m) as Element of REAL ;
deffunc H3( Nat) ->  Element of REAL = rm;
consider rs being FinSequence of REAL  such that
A26: ( len rs = m & ( for j being Nat st j in dom rs holds
rs . j = H3(j) ) ) from FINSEQ_2:sch_1();
A27: dom rs = Seg m by A26, FINSEQ_1:def_3;
 rng rs = {rm}
proof
thus  rng rs c= {rm} :: according to XBOOLE_0:def_10 ::_thesis: {rm} c= rng rs
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in rng rs or a in {rm} )
assume  a in rng rs ; ::_thesis: a in {rm}
then consider b being set  such that
A28: ( b in dom rs & a = rs . b ) by FUNCT_1:def_3;
reconsider b = b as Nat by A28;
 rs . b = rm by A28, A26;
hence  a in {rm} by A28, TARSKI:def_1; ::_thesis: verum
end;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in {rm} or a in rng rs )
assume  a in {rm} ; ::_thesis: a in rng rs
then A29: a = rm by TARSKI:def_1;
 1 <= m by NAT_1:14;
then A30: 1 in dom rs by A27;
then  a = rs . 1 by A29, A26;
hence  a in rng rs by A30, FUNCT_1:3; ::_thesis: verum
end;
then  rs = m |-> (r / (2 * m)) by A27, FUNCOP_1:9;
then A31: Sum rs = m * (r / (2 * m)) by RVSUM_1:80
.= m * ((r / 2) / m) by XCMPLX_1:78
.= r / 2 by XCMPLX_1:87 ;
now__::_thesis:_for_i_being_Element_of_NAT_st_i_in_dom_ys_holds_
ys_._i_<=_rs_._i
let i be Element of NAT ; ::_thesis: ( i in dom ys implies ys . i <= rs . i )
assume  i in dom ys ; ::_thesis: ys . i <= rs . i
then A32: i in Seg m by A22, FINSEQ_1:def_3;
then A33: ( i in dom w2 & i in dom w1 & i in dom w0 ) by A15, A16, A17, FINSEQ_1:def_3;
then consider v being Element of REAL 1 such that
A34: ( v = w2 . i & ys . i = |.v.| ) by A23;
A35: i in dom rs by A26, A32, FINSEQ_1:def_3;
reconsider p1 = partdiff (f,y1,i), p0 = partdiff (f,y0,i) as Lipschitzian LinearOperator of (REAL-NS 1),(REAL-NS 1) by LOPBAN_1:def_9;
A36: ( dom p1 = the carrier of (REAL-NS 1) & rng p1 c= the carrier of (REAL-NS 1) ) by FUNCT_2:def_1;
 <*((proj (i,m)) . t)*> in REAL 1 by FINSEQ_2:98;
then  <*((proj (i,m)) . t)*> in the carrier of (REAL-NS 1) by REAL_NS1:def_4;
then  p1 . <*((proj (i,m)) . t)*> in rng p1 by A36, FUNCT_1:3;
then reconsider P1 = p1 . <*((proj (i,m)) . t)*> as VECTOR of (REAL-NS 1) ;
A37: ( dom p0 = the carrier of (REAL-NS 1) & rng p0 c= the carrier of (REAL-NS 1) ) by FUNCT_2:def_1;
 <*((proj (i,m)) . t)*> in REAL 1 by FINSEQ_2:98;
then  <*((proj (i,m)) . t)*> in the carrier of (REAL-NS 1) by REAL_NS1:def_4;
then  p0 . <*((proj (i,m)) . t)*> in rng p0 by A37, FUNCT_1:3;
then reconsider P0 = p0 . <*((proj (i,m)) . t)*> as VECTOR of (REAL-NS 1) ;
A38: w1 /. i = w1 . i by A32, A16, PARTFUN1:def_6
.= P1 by A16, A32 ;
A39: w0 /. i = w0 . i by A32, A15, PARTFUN1:def_6
.= P0 by A15, A32 ;
A40: w2 . i = (w1 /. i) - (w0 /. i) by A33, A17
.= P1 - P0 by A39, A38, REAL_NS1:5 ;
 ( 1 <= i & i <= len S ) by A11, A32, FINSEQ_1:1;
then A41: ( s <= S . i & f is_partial_differentiable_on X,i ) by A11, A2, RFINSEQ2:2;
then  ||.(y1 - y0).|| < S . i by A12, XXREAL_0:2;
then  ||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r / (2 * m) by A10, A32, A12;
then  ||.((partdiff (f,y1,i)) - ((f `partial| (X,i)) /. y0)).|| < r / (2 * m) by A12, A41, PDIFF_1:def_20;
then A42: ||.((partdiff (f,y1,i)) - (partdiff (f,y0,i))).|| < r / (2 * m) by A7, A41, PDIFF_1:def_20;
reconsider PP = (partdiff (f,y1,i)) - (partdiff (f,y0,i)) as Lipschitzian LinearOperator of (REAL-NS 1),(REAL-NS 1) by LOPBAN_1:def_9;
A43: upper_bound (PreNorms PP) = ||.((partdiff (f,y1,i)) - (partdiff (f,y0,i))).|| by LOPBAN_1:30;
 <*((proj (i,m)) . t)*> in REAL 1 by FINSEQ_2:98;
then reconsider pt = <*((proj (i,m)) . t)*> as VECTOR of (REAL-NS 1) by REAL_NS1:def_4;
A44: PP . pt = P1 - P0 by LOPBAN_1:40;
reconsider pt1 = <*((proj (i,m)) . t)*> as Element of REAL 1 by FINSEQ_2:98;
A45: ||.pt.|| = |.pt1.| by REAL_NS1:1
.= sqrt (Sum <*(((proj (i,m)) . t) ^2)*>) by RVSUM_1:55
.= sqrt (((proj (i,m)) . t) ^2) by RVSUM_1:73 ;
A46: ((proj (i,m)) . t) ^2 >= 0
proof
percases ( (proj (i,m)) . t = 0 or (proj (i,m)) . t <> 0 ) ;
suppose (proj (i,m)) . t = 0 ; ::_thesis: ((proj (i,m)) . t) ^2 >= 0
hence  ((proj (i,m)) . t) ^2 >= 0 ; ::_thesis: verum
end;
suppose (proj (i,m)) . t <> 0 ; ::_thesis: ((proj (i,m)) . t) ^2 >= 0
hence  ((proj (i,m)) . t) ^2 >= 0 by SQUARE_1:12; ::_thesis: verum
end;
end;
end;
now__::_thesis:_not_||.pt.||_>_1
assume  ||.pt.|| > 1 ; ::_thesis: contradiction
then  ((proj (i,m)) . t) ^2 > 1 by A45, A46, SQUARE_1:18, SQUARE_1:26;
then A47: (tt . i) ^2 > 1 by PDIFF_1:def_1;
 |.tt.| <= 1 by A14, REAL_NS1:1;
then A48: Sum (sqr tt) <= 1 by SQUARE_1:18, SQUARE_1:27;
 len tt = m by CARD_1:def_7;
then  i in dom tt by A32, FINSEQ_1:def_3;
then A49: i in dom (sqr tt) by RVSUM_1:143;
now__::_thesis:_for_k_being_Element_of_NAT_st_k_in_dom_(sqr_tt)_holds_
(sqr_tt)_._k_>=_0
let k be Element of NAT ; ::_thesis: ( k in dom (sqr tt) implies (sqr tt) . b1 >= 0 )
assume  k in dom (sqr tt) ; ::_thesis: (sqr tt) . b1 >= 0
A50: (sqr tt) . k = (tt . k) ^2 by VALUED_1:11;
percases ( tt . k = 0 or tt . k <> 0 ) ;
suppose tt . k = 0 ; ::_thesis: (sqr tt) . b1 >= 0
hence  (sqr tt) . k >= 0 by A50; ::_thesis: verum
end;
suppose tt . k <> 0 ; ::_thesis: (sqr tt) . b1 >= 0
hence  (sqr tt) . k >= 0 by A50, SQUARE_1:12; ::_thesis: verum
end;
end;
end;
then  Sum (sqr tt) >= (sqr tt) . i by A49, POLYNOM5:4;
then  Sum (sqr tt) >= (tt . i) ^2 by VALUED_1:11;
hence  contradiction by A47, A48, XXREAL_0:2; ::_thesis: verum
end;
then  ( ||.(PP . pt).|| in PreNorms PP & not PreNorms PP is empty & PreNorms PP is bounded_above ) by LOPBAN_1:27;
then  ||.(PP . pt).|| <= upper_bound (PreNorms PP) by SEQ_4:def_1;
then  ||.(P1 - P0).|| <= r / (2 * m) by A44, A42, A43, XXREAL_0:2;
then  |.v.| <= r / (2 * m) by A34, A40, REAL_NS1:1;
hence  ys . i <= rs . i by A34, A26, A35; ::_thesis: verum
end;
then  Sum ys <= r / 2 by A31, A26, A22, INTEGRA5:3;
hence  mt <= r / 2 by A21, A25, XXREAL_0:2; ::_thesis: verum
end;
then A51: ( ||.((diff (f,y1)) - (diff (f,y0))).|| <= r / 2 & r / 2 < r ) by A13, A7, SEQ_4:45, XREAL_1:216;
||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| = ||.((diff (f,y1)) - ((f `| X) /. y0)).|| by A5, A12, NDIFF_1:def_9
.= ||.((diff (f,y1)) - (diff (f,y0))).|| by A5, A7, NDIFF_1:def_9 ;
hence  ||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r by A51, XXREAL_0:2; ::_thesis: verum
end;
hence  f `| X is_continuous_on X by A6, NFCONT_1:19; ::_thesis: verum
end;
assume A52: ( f is_differentiable_on X & f `| X is_continuous_on X ) ; ::_thesis: for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X )
then A53: ( X c= dom f & ( for x being Point of (REAL-NS m) st x in X holds
f is_differentiable_in x ) ) by A1, NDIFF_1:31;
thus  for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ::_thesis: verum
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m implies ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) )
assume A54: ( 1 <= i & i <= m ) ; ::_thesis: ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X )
now__::_thesis:_for_x_being_Point_of_(REAL-NS_m)_st_x_in_X_holds_
(_f_is_partial_differentiable_in_x,i_&_partdiff_(f,x,i)_=_(diff_(f,x))_*_(reproj_(i,(0._(REAL-NS_m))))_)
let x be Point of (REAL-NS m); ::_thesis: ( x in X implies ( f is_partial_differentiable_in x,i & partdiff (f,x,i) = (diff (f,x)) * (reproj (i,(0. (REAL-NS m)))) ) )
assume  x in X ; ::_thesis: ( f is_partial_differentiable_in x,i & partdiff (f,x,i) = (diff (f,x)) * (reproj (i,(0. (REAL-NS m)))) )
then  f is_differentiable_in x by A52, A1, NDIFF_1:31;
hence  ( f is_partial_differentiable_in x,i & partdiff (f,x,i) = (diff (f,x)) * (reproj (i,(0. (REAL-NS m)))) ) by A54, Th21; ::_thesis: verum
end;
then  for x being Point of (REAL-NS m) st x in X holds
f is_partial_differentiable_in x,i ;
hence A55: f is_partial_differentiable_on X,i by A1, A54, Th8, A53; ::_thesis: f `partial| (X,i) is_continuous_on X
then A56: dom (f `partial| (X,i)) = X by PDIFF_1:def_20;
 for y0 being Point of (REAL-NS m)
for r being Real st y0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r ) )
proof
let y0 be Point of (REAL-NS m); ::_thesis: for r being Real st y0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r ) )
let r be Real; ::_thesis: ( y0 in X & 0 < r implies ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r ) ) )
assume A57: ( y0 in X & 0 < r ) ; ::_thesis: ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r ) )
then consider s being Real such that
A58: ( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r ) ) by A52, NFCONT_1:19;
take  s ; ::_thesis: ( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r ) )
thus  0 < s by A58; ::_thesis: for y1 being Point of (REAL-NS m) st y1 in X & ||.(y1 - y0).|| < s holds
||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r
let y1 be Point of (REAL-NS m); ::_thesis: ( y1 in X & ||.(y1 - y0).|| < s implies ||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r )
assume A59: ( y1 in X & ||.(y1 - y0).|| < s ) ; ::_thesis: ||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r
then  ||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r by A58;
then  ||.((diff (f,y1)) - ((f `| X) /. y0)).|| < r by A59, A52, NDIFF_1:def_9;
then A60: ||.((diff (f,y1)) - (diff (f,y0))).|| < r by A57, A52, NDIFF_1:def_9;
 ( f is_differentiable_in y1 & f is_differentiable_in y0 ) by A52, A1, A59, A57, NDIFF_1:31;
then A61: ( partdiff (f,y1,i) = (diff (f,y1)) * (reproj (i,(0. (REAL-NS m)))) & partdiff (f,y0,i) = (diff (f,y0)) * (reproj (i,(0. (REAL-NS m)))) ) by Th21, A54;
reconsider PP = (partdiff (f,y1,i)) - (partdiff (f,y0,i)) as Lipschitzian LinearOperator of (REAL-NS 1),(REAL-NS 1) by LOPBAN_1:def_9;
reconsider DD = (diff (f,y1)) - (diff (f,y0)) as Lipschitzian LinearOperator of (REAL-NS m),(REAL-NS 1) by LOPBAN_1:def_9;
A62: upper_bound (PreNorms PP) = ||.((partdiff (f,y1,i)) - (partdiff (f,y0,i))).|| by LOPBAN_1:30;
A63: upper_bound (PreNorms DD) = ||.((diff (f,y1)) - (diff (f,y0))).|| by LOPBAN_1:30;
A64: ( PreNorms PP is bounded_above & PreNorms DD is bounded_above ) by LOPBAN_1:28;
 PreNorms PP c= PreNorms DD
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in PreNorms PP or a in PreNorms DD )
assume  a in PreNorms PP ; ::_thesis: a in PreNorms DD
then consider t being VECTOR of (REAL-NS 1) such that
A65: ( a = ||.(PP . t).|| & ||.t.|| <= 1 ) ;
A66: dom (reproj (i,(0. (REAL-NS m)))) = the carrier of (REAL-NS 1) by FUNCT_2:def_1;
set tm = (reproj (i,(0. (REAL-NS m)))) . t;
A67: ||.((reproj (i,(0. (REAL-NS m)))) . t).|| <= 1 by A65, Th5, A54;
A68: (partdiff (f,y1,i)) . t = (diff (f,y1)) . ((reproj (i,(0. (REAL-NS m)))) . t) by A66, A61, FUNCT_1:13;
 (partdiff (f,y0,i)) . t = (diff (f,y0)) . ((reproj (i,(0. (REAL-NS m)))) . t) by A66, A61, FUNCT_1:13;
then ||.(PP . t).|| = ||.(((diff (f,y1)) . ((reproj (i,(0. (REAL-NS m)))) . t)) - ((diff (f,y0)) . ((reproj (i,(0. (REAL-NS m)))) . t))).|| by A68, LOPBAN_1:40
.= ||.(DD . ((reproj (i,(0. (REAL-NS m)))) . t)).|| by LOPBAN_1:40 ;
hence  a in PreNorms DD by A65, A67; ::_thesis: verum
end;
then  ||.((partdiff (f,y1,i)) - (partdiff (f,y0,i))).|| <= ||.((diff (f,y1)) - (diff (f,y0))).|| by A64, A62, A63, SEQ_4:48;
then  ||.((partdiff (f,y1,i)) - (partdiff (f,y0,i))).|| < r by A60, XXREAL_0:2;
then  ||.((partdiff (f,y1,i)) - ((f `partial| (X,i)) /. y0)).|| < r by A57, A55, PDIFF_1:def_20;
hence  ||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r by A59, A55, PDIFF_1:def_20; ::_thesis: verum
end;
hence  f `partial| (X,i) is_continuous_on X by A56, NFCONT_1:19; ::_thesis: verum
end;
end;