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 Th14:
for x,y be Point of REAL-NS 1,i be Nat st 1 <=i & i <= m holds
  reproj(i,0.(REAL-NS m)).(x+y)
    = reproj(i,0.(REAL-NS m)).x+reproj(i,0.(REAL-NS m)).y
proof
   let x,y be Point of REAL-NS 1,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 q2 be Element of REAL, z2 be Element of REAL m such that
A3: y = <*q2*> & z2 = 0.(REAL-NS m)
    & reproj(i,0.(REAL-NS m)).y = reproj(i,z2).q2 by PDIFF_1:def 6;
   consider q12 be Element of REAL, z12 be Element of REAL m such that
A4:x+y = <*q12*> & z12 = 0.(REAL-NS m)
    & reproj(i,0.(REAL-NS m)).(x+y) = reproj(i,z12).q12 by PDIFF_1:def 6;
A5:0.(REAL-NS m) = 0*m by REAL_NS1:def 4;
   reconsider qq1= <*q1*> as Element of REAL 1 by FINSEQ_2:98;
   reconsider qq2= <*q2*> as Element of REAL 1 by FINSEQ_2:98;
    x+y = qq1 + qq2 by A2,A3,REAL_NS1:2;
then A6:x+y = <*q1+q2*> by RVSUM_1:13;
    reproj(i,0.(REAL-NS m)).x+reproj(i,0.(REAL-NS m)).y
     = reproj(i,(0*m)).q1 + reproj(i,(0*m)).q2 by A2,A3,A5,REAL_NS1:2
    .= reproj(i,(0*m)).(q1+q2) by A1,Th13;
   hence thesis by A6,A4,A5,FINSEQ_1:76;
end;
