reserve n,m,i,k for Element of NAT;
reserve x,X,X1 for set;
reserve r,p for Real;
reserve s,x0,x1,x2 for Real;
reserve f,f1,f2 for PartFunc of REAL,REAL n;
reserve h for PartFunc of REAL,REAL-NS n;
reserve W for non empty set;

theorem Th43:
for n be non zero Element of NAT, h be PartFunc of REAL, REAL n holds
h is_continuous_in x0 iff
    (x0 in dom h & for i be Element of NAT st i in Seg n
    holds proj(i,n)*h is_continuous_in x0)
proof
let n be non zero Element of NAT, h be PartFunc of REAL,REAL n;
hereby
 assume A1: h is_continuous_in x0;
  hence
  A2: x0 in dom h by Th3;
  thus for i be Element of NAT st i in Seg n
    holds proj(i,n)*h is_continuous_in x0
  proof
   let i be Element of NAT;
   assume i in Seg n; then
   A3: 1<=i & i <= n by FINSEQ_1:1;
   A4:dom (proj(i,n)) = REAL n by FUNCT_2:def 1;
   rng h c= REAL n; then
A5: dom (proj(i,n)*h) = dom h by A4,RELAT_1:27;
   proj(i,n) is_continuous_in h/.x0 by A3,Th42;
   hence thesis by A5,A1,A2,Th22;
  end;
end;
assume
A6: x0 in dom h &
   for i be Element of NAT st i in Seg n
      holds (proj(i,n)*h) is_continuous_in x0;
for r be Real st 0<r ex s be Real
st 0<s & for x1 be Real st x1
  in dom h & |.x1-x0.|<s holds |. h/.x1 - h/.x0 .|<r
proof
 let r0 be Real;
 assume A7: 0 < r0;
 set r=r0/2;
 A8:0 < r by A7,XREAL_1:215;
  defpred P[Nat,Real] means 0 < $2 &
  for x1 be Real st x1 in dom h & |.x1-x0.| < $2 holds
  |. (proj($1,n)*h).x1 - (proj($1,n)*h).x0  .| < r/n;
  A9: 0 < r/n by A8,XREAL_1:139;
  A10: for j be Nat st j in Seg n ex x be Element of REAL st P[j,x]
     proof
      let j be Nat;
      assume A11: j in Seg n;
  A12: proj(j,n)*h is_continuous_in x0 by A6,A11;
     A13:dom (proj(j,n)) = REAL n by FUNCT_2:def 1;
     rng h c= REAL n; then
     A14:dom (proj(j,n)*h) = dom h by A13,RELAT_1:27;
    consider sj be Real such that
   A15: 0 < sj &
      for x1 be Real st x1 in dom (proj(j,n)*h) & |.x1-x0.| < sj holds
       |.(proj(j,n)*h).x1 - (proj(j,n)*h).x0 .| < r/n by A12,A9,FCONT_1:3;
    sj in REAL by XREAL_0:def 1;
    hence thesis by A15,A14;
  end;
consider s0 be FinSequence of REAL such that
A16: dom s0 = Seg n &
     for k be Nat st k in Seg n holds P[k,s0.k] from FINSEQ_1:sch 5(A10);
     n in Seg n by FINSEQ_1:3;
 then reconsider rs0= rng s0 as non empty ext-real-membered set
      by A16,FUNCT_1:3;
 rng s0 is finite by A16,FINSET_1:8;
 then reconsider rs0 as left_end right_end non empty ext-real-membered set;
A17:min rs0 in rng s0 by XXREAL_2:def 7;
     reconsider s = min rs0 as Real;
     take s;
     ex i1 be object st i1 in dom s0 & s = s0.i1 by A17,FUNCT_1:def 3;
     hence 0 < s by A16;
now let x1;
  assume A18: x1 in dom h & |.x1- x0.| < s;
 now let j be Element of NAT;
   assume 1<=j & j <= n; then
A19: j in Seg n by FINSEQ_1:1;
    then s0.j in rng s0 by A16,FUNCT_1:3;
    then s <= s0.j by XXREAL_2:def 7;
    then |.x1-x0.| < s0.j by A18,XXREAL_0:2; then
A20: |. (proj(j,n)*h).x1 - (proj(j,n)*h).x0 .| < r/n by A19,A18,A16;
    A21:(proj(j,n)*h).x1 =(proj(j,n)).(h.x1) by A18,FUNCT_1:13
         .=(proj(j,n)).(h/.x1) by A18,PARTFUN1:def 6;
    (proj(j,n)*h).x0 = (proj(j,n)).(h.x0) by A6,FUNCT_1:13
    .=(proj(j,n)).(h/.x0) by A6,PARTFUN1:def 6;
   hence |. proj(j,n).((h/.x1) - (h/.x0)) .| <= r/n by A20,A21,PDIFF_8:12;
 end;
 then |. h/.x1 - h/.x0 .| <= n*(r/n) by PDIFF_8:17; then
A22:  |. h/.x1 - h/.x0 .| <= r by XCMPLX_1:87;
  r < r0 by A7,XREAL_1:216;
  hence|. h/.x1 - h/.x0 .| < r0 by A22,XXREAL_0:2;
  end;
  hence for x1 st x1 in dom h & |.x1-x0.| < s
    holds |. h/.x1 - h/.x0 .| < r0;
 end;
hence thesis by A6,Th3;
end;
