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 Th46:
for n be non zero Element of NAT,
    h be PartFunc of REAL,REAL-NS n holds
 h is_continuous_in x0 iff
  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-NS n;
hereby assume A1: h is_continuous_in x0;
  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
   A2: 1<=i & i <= n by FINSEQ_1:1;
   A3:dom (Proj(i,n)) = the carrier of REAL-NS n by FUNCT_2:def 1;
   rng h c= the carrier of REAL-NS n; then
A4: dom (Proj(i,n)*h) = dom h by A3,RELAT_1:27;
   Proj(i,n) is_continuous_in h/.x0 by A2,Th45;
   hence (Proj(i,n)*h) is_continuous_in x0 by A4,A1,NFCONT_3:15;
  end;
end;
assume A5: for i be Element of NAT st i in Seg n
    holds (Proj(i,n)*h) is_continuous_in x0;
1<=n by NAT_1:14;
then 1 in Seg n by FINSEQ_1:1;
then (Proj(1,n)*h) is_continuous_in x0 by A5; then
A6:x0 in dom(Proj(1,n)*h);
A7:dom(Proj(1,n)*h) c= dom h by RELAT_1:25;
for r being Real st 0<r ex s st 0<s & for x1 st x1
  in dom h & |.x1-x0.|<s holds ||. h/.x1 - h/.x0 .||<r
proof
 let r0 be Real;
 set r=r0/2;
 assume A8: 0 < r0; then
 A9:0 < r by XREAL_1:215;
  defpred P[set,Real] means
  ex j be Element of NAT st $1=j & 0 < $2 &
  for x1 st x1 in dom h & |.x1-x0.| < $2 holds
      ||. (Proj(j,n)*h)/.x1 - (Proj(j,n)*h)/.x0 .|| < r/n;
  A10: 0 < r/n by A9,XREAL_1:139;
  A11: for j0 be Nat st j0 in Seg n holds
     ex x be Element of REAL st P[j0,x]
     proof
      let j0 be Nat;
      assume A12: j0 in Seg n;
      reconsider j=j0 as Element of NAT by ORDINAL1:def 12;
     A13: Proj(j,n)*h is_continuous_in x0 by A5,A12;
     A14:dom (Proj(j,n)) = the carrier of REAL-NS n by FUNCT_2:def 1;
     rng h c= the carrier of REAL-NS n; then
     A15:dom (Proj(j,n)*h) = dom h by A14,RELAT_1:27;
    consider sj be Real such that
A16: 0 < sj &
     for x1 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 A13,A10,NFCONT_3:8;
     sj in REAL by XREAL_0:def 1;
   hence thesis by A16,A15;
  end;
consider s0 be FinSequence of REAL such that
A17: dom s0 = Seg n &
     for k be Nat st k in Seg n holds P[k,s0.k] from FINSEQ_1:sch 5(A11);
     n in Seg n by FINSEQ_1:3;
 then reconsider rs0= rng s0 as non empty ext-real-membered set
      by A17,FUNCT_1:3;
 rng s0 is finite by A17,FINSET_1:8;
 then reconsider rs0 as left_end right_end non empty ext-real-membered set;
A18:min rs0 in rng s0 by XXREAL_2:def 7;
    reconsider s = min rs0 as Real;
   take s;
   consider i1 be object such that
A19: i1 in dom s0 & s = s0.i1 by A18,FUNCT_1:def 3;
   ex j be Element of NAT st i1=j & 0 < s0.i1 &
      for x1 st x1 in dom h & |.x1-x0.| < s0.i1 holds
       ||. (Proj(j,n)*h)/.x1 - (Proj(j,n)*h)/.x0 .|| < r/n by A17,A19;
  hence 0 < s by A19;
  let x1;
  assume A20: x1 in dom h & |.x1-x0.| < s;
 now let j be Element of NAT;
 assume 1<=j & j <= n; then
     A21:j in Seg n by FINSEQ_1:1;
     then consider jj be Element of NAT such that
     A22: jj = j & 0 < s0.j &
      for x1 st x1 in dom h & |.x1-x0.|< s0.j holds
    ||. (Proj(jj,n)*h)/.x1 - (Proj(jj,n)*h)/.x0 .|| < r/n by A17;
    s0.j in rng s0 by A21,A17,FUNCT_1:3;
    then s <= s0.j by XXREAL_2:def 7;
    then |.x1-x0.| < s0.j by A20,XXREAL_0:2; then
   A23: ||. (Proj(j,n)*h)/.x1 - (Proj(j,n)*h)/.x0 .|| < r/n by A22,A20;
    A24:dom (Proj(j,n)) = the carrier of REAL-NS n by FUNCT_2:def 1; then
    A25:(Proj(j,n)*h)/.x1 = (Proj(j,n))/.(h/.x1) by A20,PARTFUN2:4;
   (Proj(j,n)*h)/.x0 = (Proj(j,n))/.(h/.x0) by A24,A7,A6,PARTFUN2:4;
   hence ||. Proj(j,n).((h/.x1) - (h/.x0)) .|| <= r/n
                                             by A23,A25,PDIFF_8:11;
 end;
 then ||. h/.x1 - h/.x0 .|| <= n*(r/n) by PDIFF_8:16; then
A26: ||. h/.x1 - h/.x0 .|| <= r by XCMPLX_1:87;
  r < r0 by A8,XREAL_1:216;
  hence ||. h/.x1 - h/.x0 .|| < r0 by A26,XXREAL_0:2;
end;
hence thesis by A7,A6,NFCONT_3:8;
end;
