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

theorem Th73:
for f be PartFunc of REAL m,REAL, I,G be non empty FinSequence of NAT
 st f is_partial_differentiable_on Z,G
holds
 for n be Nat st n <= len I holds
   (PartDiffSeq((f`partial|(Z,G)),Z,I)).n
     = (PartDiffSeq(f,Z,(G^I))).(len G + n)
proof
   let f be PartFunc of REAL m,REAL, I,G be non empty FinSequence of NAT;
   assume A1:f is_partial_differentiable_on Z,G;
   set g = f`partial|(Z,G);
A2:dom G c= dom (G^I) by FINSEQ_1:26;
A3:for i be Nat st i <= len G-1  holds (G^I)/.(i+1) = G/.(i+1)
   proof
    let i be Nat;
    assume i <= len G - 1; then
    1 <= i+1 & i+1 <= len G by NAT_1:11,XREAL_1:19; then
A4: i+1 in dom G by FINSEQ_3:25; then
    (G^I)/.(i+1) = (G^I).(i+1) by A2,PARTFUN1:def 6; then
    (G^I)/.(i+1) = G.(i+1) by A4,FINSEQ_1:def 7;
    hence (G^I)/.(i+1) = G/.(i+1) by A4,PARTFUN1:def 6;
   end;
A5:len (G^I) = len G + len I by FINSEQ_1:22;
A6:for i be Nat st i <= (len I)-1 holds (G^I)/.(len G + (i+1)) = I/.(i+1)
   proof
    let i be Nat;
    assume i <= len I - 1; then
A7: i+1 <= len I by XREAL_1:19; then
A8: i+1 in dom I by NAT_1:11,FINSEQ_3:25;
    1 <= len G + (i+1) by NAT_1:11,XREAL_1:38; then
    len G + (i+1) in dom (G^I) by A5,A7,XREAL_1:7,FINSEQ_3:25;
    hence (G^I)/.(len G + (i+1)) =(G^I).(len G + (i+1)) by PARTFUN1:def 6
            .= I.(i+1) by A8,FINSEQ_1:def 7
            .= I/.(i+1) by A8,PARTFUN1:def 6;
   end;
   defpred P0[Nat] means
    $1 <= len G implies
     (PartDiffSeq(f,Z,G^I)).$1 =(PartDiffSeq(f,Z,G)).$1;
A9:P0[0]
   proof
    assume 0 <= len G;
    (PartDiffSeq(f,Z,G^I)).0 = f |Z  by Def7;
    hence (PartDiffSeq(f,Z,G^I)).0 = (PartDiffSeq(f,Z,G)).0 by Def7;
   end;
A10: for k be Nat st P0[k] holds P0[k+1]
   proof
    let k be Nat;
    assume A11: P0[k];
    assume A12: k+1 <= len G; then
A122:    k +1 -1 <= len G -1 by XREAL_1:9;
    k <= k+1 by NAT_1:11; then
A13: k <= len G by A12,XXREAL_0:2;
    thus (PartDiffSeq(f,Z,G^I)).(k+1)
        = ((PartDiffSeq(f,Z,G^I)).k)`partial|(Z,(G^I)/.(k+1)) by Def7
       .= ((PartDiffSeq(f,Z,G)).k)`partial|(Z,G/.(k+1))
             by A13,A3,A11,A122
       .= (PartDiffSeq(f,Z,G)).(k+1) by Def7;
   end;
   for n be Nat holds P0[n] from NAT_1:sch 2(A9,A10); then
A15:(PartDiffSeq(f,Z,G^I)).(len G) = (PartDiffSeq(f,Z,G)).(len G);
   defpred P[Nat] means
    $1 <= len I implies
      (PartDiffSeq(g,Z,I)).$1 =(PartDiffSeq(f,Z,(G^I))).(len G + $1);
A16:
   P[0]
   proof
    assume 0 <= len I;
    (PartDiffSeq(f,Z,(G^I))).(len G + 0) = g|Z by A1,A15,Th72;
    hence thesis by Def7;
   end;
A17:
   for k be Nat st P[k] holds P[k+1]
   proof
    let k be Nat;
    assume A18: P[k];
    set i = (len G) + k;
    assume A19: k+1 <= len I; then
    A199: k +1 -1 <= (len I) -1 by XREAL_1:9;
A20:k <= k+1 by NAT_1:11;
    (G^I)/.(i+1) = (G^I)/.(len G + (k+1)); then
A21:(G^I)/.(i+1) = I/.(k+1) by A6,A199;
    (PartDiffSeq(f,Z,(G^I))).(len G + (k+1))
      = ((PartDiffSeq(f,Z,(G^I))).i)`partial|(Z,(G^I)/.(i+1)) by Def7;
    hence thesis by A20,A19,A18,A21,Def7,XXREAL_0:2;
   end;
   for n be Nat holds P[ n ] from NAT_1:sch 2(A16,A17);
   hence thesis;
end;
