reserve m,n for non zero Element of NAT;
reserve i,j,k for Element of NAT;
reserve Z for set;

theorem Th78:
for X be Subset of REAL m, r be Real, I be non empty FinSequence of NAT,
    f be PartFunc of REAL m,REAL st
  X is open & rng I c= Seg m & f is_partial_differentiable_on X,I
holds
   for i st i <= (len I)-1 holds
     (PartDiffSeq(r(#)f,X,I)).i is_partial_differentiable_on X,I/.(i+1)
   & (PartDiffSeq(r(#)f,X,I)).i = r(#)(PartDiffSeq(f,X,I).i)
proof
   let Z be Subset of REAL m, r be Real, I be non empty FinSequence of NAT,
       f be PartFunc of REAL m,REAL;
   assume
A1: Z is open & rng I c= Seg m
        & f is_partial_differentiable_on Z,I;
   defpred P[Nat] means
    $1 <= (len I)-1 implies
     ( (PartDiffSeq(r(#)f,Z,I)).$1 is_partial_differentiable_on Z,I/.($1+1)
     & (PartDiffSeq(r(#)f,Z,I)).$1 = r(#)((PartDiffSeq(f,Z,I)).$1) );
   reconsider Z0=0 as Element of NAT;
A2:P[0]
   proof
    assume 0 <= (len I)-1; then
A3: (PartDiffSeq(f,Z,I)).Z0
     is_partial_differentiable_on Z,I/.( Z0 + 1) by A1;
A4:   (r(#)f) |Z = r(#)(f|Z) by RFUNCT_1:49;
    (PartDiffSeq(r(#)f,Z,I)).Z0 = (r(#)f) | Z by Def7; then
A5: (PartDiffSeq(r(#)f,Z,I)).Z0 = r(#)((PartDiffSeq(f,Z,I)).Z0) by A4,Def7;
    1 <= len I by FINSEQ_1:20; then
    I/.1 in Seg m by A1,Lm6; then
    1<=I/.1 & I/.1 <= m by FINSEQ_1:1;
    hence thesis by A5,A1,A3,Th67;
   end;
A6:for k be Nat st P[k] holds P[k+1]
   proof
    let k be Nat;
    assume A7: P[k];
    assume A8: k+1 <=(len I)-1;
A9: k <= k+1 by NAT_1:11; then
A10:k <=(len I)-1 by A8,XXREAL_0:2;
A11:(PartDiffSeq(f,Z,I)).(k+1)
      is_partial_differentiable_on Z,I/.((k+1)+1) by A8,A1;
    k+1 <= (len I)-1 + 1 by A10,XREAL_1:6; then
    I/.(k+1) in Seg m by A1,Lm6,NAT_1:11; then
A12: 1<=I/.(k+1) & I/.(k+1) <= m by FINSEQ_1:1;
   k in NAT by ORDINAL1:def 12; then
A13:(PartDiffSeq(f,Z,I)).k
       is_partial_differentiable_on Z,I/.(k+1) by A1,A9,A8,XXREAL_0:2;
    (k+1)+1 <=(len I)-1 +1 by A8,XREAL_1:6; then
    I/.((k+1)+1) in Seg m by A1,Lm6,NAT_1:11; then
A14: 1<=I/.((k+1)+1) & I/.((k+1)+1) <= m by FINSEQ_1:1;
A15:(PartDiffSeq(r(#)f,Z,I)).(k+1)
      = (r(#)(PartDiffSeq(f,Z,I)).k )`partial|(Z,I/.(k+1))
          by A9,A8,A7,Def7,XXREAL_0:2
     .= r(#)(((PartDiffSeq(f,Z,I)).k)`partial|(Z,I/.(k+1)))
          by A13,A1,A12,Th67
     .= r(#)((PartDiffSeq(f,Z,I)).(k+1)) by Def7;
    hence (PartDiffSeq(r(#)f,Z,I)).(k+1)
        is_partial_differentiable_on Z,I/.((k+1)+1) by A1,A11,A14,Th67;
    thus (PartDiffSeq(r(#)f,Z,I)).(k+1)
        = r(#)(PartDiffSeq(f,Z,I)).(k+1) by A15;
   end;
   for n be Nat holds P[ n ] from NAT_1:sch 2(A2,A6);
   hence thesis;
end;
