reserve n for Nat,
        p,p1,p2 for Point of TOP-REAL n,
        x for Real;
reserve n,m for non zero Nat;
reserve i,j for Nat;
reserve f for PartFunc of REAL-NS m,REAL-NS n;
reserve g for PartFunc of REAL m,REAL n;
reserve h for PartFunc of REAL m,REAL;
reserve x for Point of REAL-NS m;
reserve y for Element of REAL m;
reserve X for set;

theorem Th41:
for m be non zero Nat, f be PartFunc of REAL m,REAL,
    g be PartFunc of REAL ,REAL,
    x,y be Element of REAL m,
    i be Nat,
    xi be Real st
1 <=i & i <= m & y=reproj(i,x).xi & g=f*reproj(i,x)
    holds diff(g,xi) = partdiff(f,y,i)
proof
   let m be non zero Nat,
       f be PartFunc of REAL m,REAL,
       g be PartFunc of REAL ,REAL,
       x,y be Element of REAL m,
       i be Nat,
       xi be Real;
   assume A1: 1 <=i & i <= m & y=reproj(i,x).xi & g=f*reproj(i,x);
   then reproj(i,x)=reproj(i,y) & proj(i,m).y=xi by Th39,Th40;
   hence partdiff(f,y,i) = diff(g,xi) by A1;
end;
