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

theorem Th74:
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;
   let 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;
   thus for i be Element of NAT st i <= (len I)-1 holds
      (PartDiffSeq(f+g,Z,I)).i is_partial_differentiable_on Z,I/.(i+1)
    & (PartDiffSeq(f+g,Z,I)).i = PartDiffSeq(f,Z,I).i + PartDiffSeq(g,Z,I).i
   proof
    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 = (PartDiffSeq(f,Z,I)).Z0 &
     g|Z = (PartDiffSeq(g,Z,I)).Z0
     & (PartDiffSeq(f+g,Z,I)).Z0 = (f+g) |Z by Def7;
A5:   f|Z + g|Z = (f+g) |Z by RFUNCT_1:44;
     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 A4,A5,A1,A3,Th65;
    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,Z,I)).k + (PartDiffSeq(g,Z,I)).k)`partial|(Z,I/.(k+1))
           by A9,A7,A8,Def7,XXREAL_0:2
      .= ((PartDiffSeq(f,Z,I)).k)`partial|(Z,I/.(k+1))
        + ((PartDiffSeq(g,Z,I)).k)`partial|(Z,I/.(k+1)) by A13,A1,A12,Th65
      .= (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,Th65;
     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;
end;
