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 Th16:
for x be Point of REAL-NS 1, a be Real, i be Nat st
 1 <=i & i <= m holds
   reproj(i,0.(REAL-NS m)).(a*x) = a*(reproj(i,0.(REAL-NS m)).x)
proof
   let x be Point of REAL-NS 1,a be Real,
       i be Nat;
   assume A1: 1 <=i & i <= m;
   consider q1 be Element of REAL, z1 be Element of REAL m such that
A2: x = <*q1*> & z1 = 0.(REAL-NS m)
    & reproj(i,0.(REAL-NS m)).x = reproj(i,z1).q1 by PDIFF_1:def 6;
   consider q12 be Element of REAL, z12 be Element of REAL m such that
A3:a*x = <*q12*> & z12 = 0.(REAL-NS m)
    & reproj(i,0.(REAL-NS m)).(a*x) = reproj(i,z12).q12 by PDIFF_1:def 6;
A4:0.(REAL-NS m) = 0*m by REAL_NS1:def 4;
   reconsider qq1= <*q1*> as  Element of REAL 1 by FINSEQ_2:98;
    a*x = a*qq1 by A2,REAL_NS1:3;
then A5:a*x = <*a*q1*> by RVSUM_1:47;
    a*(reproj(i,0.(REAL-NS m)).x) = a*(reproj(i,0*m).q1) by A2,A4,REAL_NS1:3
    .= reproj(i,0*m).(a*q1) by A1,Th15;
   hence thesis by A5,A3,A4,FINSEQ_1:76;
end;
