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

theorem Th76:
for X be Subset of REAL m, I be non empty FinSequence of NAT,
    f,g be PartFunc of REAL m,REAL st
  X is open & rng I c= Seg m
  & f is_partial_differentiable_on X,I
  & g is_partial_differentiable_on X,I
holds
  for i st i <= (len I)-1 holds
   (PartDiffSeq(f-g,X,I)).i is_partial_differentiable_on X,I/.(i+1)
 & (PartDiffSeq(f-g,X,I)).i = (PartDiffSeq(f,X,I).i)-(PartDiffSeq(g,X,I).i)
proof
   let Z be Subset of REAL m, I be non empty FinSequence of NAT,
       f,g be PartFunc of REAL m,REAL;
   assume A1: Z is open & rng I c= Seg m
       & f is_partial_differentiable_on Z,I
       & g is_partial_differentiable_on Z,I;
   defpred P[Nat] means
     $1 <= (len I)-1 implies
     (  (PartDiffSeq(f-g,Z,I)).$1 is_partial_differentiable_on Z,I/.($1+1)
      & (PartDiffSeq(f-g,Z,I)).$1
         = (PartDiffSeq(f,Z,I)).$1 - (PartDiffSeq(g,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) &
    (PartDiffSeq(g,Z,I)).Z0 is_partial_differentiable_on Z,I/.(Z0 + 1)
       by A1;
A4: f|Z - g|Z = (f-g) |Z by RFUNCT_1:47;
A5: f|Z = (PartDiffSeq(f,Z,I)).0 & g|Z = (PartDiffSeq(g,Z,I)).0
  & (f-g) |Z = (PartDiffSeq(f-g,Z,I)).0 by 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,A4,A1,A3,Th66;
   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) &
    (PartDiffSeq(g,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) &
    (PartDiffSeq(g,Z,I)).k is_partial_differentiable_on Z,I/.(k+1)
        by A9,A1,A8,XXREAL_0:2;
A14: (PartDiffSeq(f,Z,I)).(k+1)
       = ((PartDiffSeq(f,Z,I)).k)`partial|(Z,I/.(k+1)) by Def7;
    (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
A15: 1<=I/.((k+1)+1) & I/.((k+1)+1) <= m by FINSEQ_1:1;
A16:(PartDiffSeq(f-g,Z,I)).(k+1)
        = ((PartDiffSeq(f-g,Z,I)).k)`partial|(Z,I/.(k+1)) by Def7
       .= ((PartDiffSeq(f,Z,I)).k)`partial|(Z,I/.(k+1))
            - ((PartDiffSeq(g,Z,I)).k)`partial|(Z,I/.(k+1))
              by A13,A1,A12,Th66,A9,A7,A8,XXREAL_0:2
       .= (PartDiffSeq(f,Z,I)).(k+1) - (PartDiffSeq(g,Z,I)).(k+1)
              by A14,Def7;
    hence
     (PartDiffSeq(f-g,Z,I)).(k+1) is_partial_differentiable_on Z,I/.((k+1)+1)
        by A1,A11,A15,Th66;
    thus (PartDiffSeq(f-g,Z,I)).(k+1)
        = (PartDiffSeq(f,Z,I)).(k+1) - (PartDiffSeq(g,Z,I)).(k+1) by A16;
   end;
   for n be Nat holds P[ n ] from NAT_1:sch 2(A2,A6);
   hence thesis;
end;
