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;
