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 Th20:
for x,y  be Point of REAL-NS 1,i be Nat, z be Point of REAL-NS m st
 1 <=i & i <= m & y=Proj(i,m).z holds
   reproj(i,z).x - z = reproj(i,0.(REAL-NS m)).(x-y)
   & z - reproj(i,z).x = reproj(i,0.(REAL-NS m)).(y-x)
proof
   let x,y be Point of REAL-NS 1, i be Nat,
       z be Point of REAL-NS m;
   assume A1: 1 <=i & i <= m & y=Proj(i,m).z;
   consider q1 be Element of REAL, z1 be Element of REAL m such that
A2: x = <*q1*> & z1 = z & reproj(i,z).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 = z & reproj(i,z).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;
   consider q21 be Element of REAL, z21 be Element of REAL m such that
A5:y-x = <*q21*> & z21 = 0.(REAL-NS m)
    & reproj(i,0.(REAL-NS m)).(y-x) = reproj(i,z21).q21 by PDIFF_1:def 6;
A6:0.(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 & y-x = qq2 - qq1 by A2,A3,REAL_NS1:5;
   then x-y =<*q1-q2*> & y-x = <*q2-q1*> by RVSUM_1:29;
then A7:reproj(i,0.(REAL-NS m)).(x-y)=reproj(i,(0*m)).(q1-q2)
   & reproj(i,0.(REAL-NS m)).(y-x)=reproj(i,(0*m)).(q2-q1)
      by A4,A5,A6,FINSEQ_1:76;
    y = <* proj(i,m).z *> by A1,PDIFF_1:def 4;
   then q2=proj(i,m).z1 by A2,A3,FINSEQ_1:76;
   then reproj(i,z1).q1 - z1 = reproj(i,(0*m)).(q1-q2)
   & z1 - reproj(i,z1).q1 = reproj(i,0*m).(q2-q1) by Th19,A1;
   hence thesis by A7,A2,REAL_NS1:5;
end;
