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 Th13:
for x,y be Element of REAL,i be Nat st 1 <=i & i <= m holds
  reproj(i,0*m).(x+y) = reproj(i,0*m).x+reproj(i,0*m).y
proof
   let x,y be Element of REAL,i be Nat;
   assume A1: 1 <=i & i <= m;
    reconsider xy = x+y as Element of REAL;
Replace(0*m,i,x) = reproj(i,0*m).x & Replace(0*m,i,y) = reproj(i,0*m).y &
   (reproj(i,0*m)).(x+y) = Replace(0*m,i,xy) by PDIFF_1:def 5;
   hence thesis by A1,Th9;
end;
