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 Th15:
for x,a be Real,i be Nat st 1 <= i <= m holds
  reproj(i,0*m).(a*x) = a(#)(reproj(i,0*m).x)
proof
   let x,a be Real,i be Nat;
   assume
A1:   1 <=i & i <= m;
    reconsider a,x as Element of REAL by XREAL_0:def 1;
    reconsider ax = a*x as Element of REAL;
A2:Replace(0*m,i,ax) = a*(Replace(0*m,i,x)) by Th10,A1;
    Replace(0*m,i,x) = reproj(i,0*m).x by PDIFF_1:def 5;
  then (reproj(i,0*m)).(a*x) = a*(reproj(i,0*m).x) by A2,PDIFF_1:def 5;
 hence thesis;
end;
