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 Th7:
for m,i be Nat, x be Point of REAL-NS m,
  p be Point of REAL-NS 1 holds
  (reproj(i,x)).p - x = reproj(i,0.(REAL-NS m)).(p - Proj(i,m).x) &
  x - (reproj(i,x)).p = reproj(i,0.(REAL-NS m)).(Proj(i,m).x - p)
proof
   let m,i be Nat,
       x be Point of REAL-NS m, p be Point of REAL-NS 1;
   consider p1 be Element of REAL, y be Element of REAL m such that
A1: p = <* p1 *> & y = x & (reproj(i,x)).p = reproj(i,y).p1 by PDIFF_1:def 6;
   reconsider pm = p as Element of REAL 1 by REAL_NS1:def 4;
   reconsider rm = reproj(i,y).p1 as Element of REAL m;
    (reproj(i,x)).p - x = rm - y by A1,REAL_NS1:5;
then A2:(reproj(i,x)).p - x = reproj(i,0*m).(p1 - proj(i,m).y) by Th6;
A3:0*m = 0.(REAL-NS m) by REAL_NS1:def 4;
A4:<* proj(i,m).y *> = Proj(i,m).x by A1,PDIFF_1:def 4;
   reconsider Pr = Proj(i,m).x as Element of REAL 1 by REAL_NS1:def 4;
   consider r1 be Element of REAL, z be Element of REAL m such that
A5: p - Proj(i,m).x = <* r1 *> & z = 0.(REAL-NS m)
    & (reproj(i,0.(REAL-NS m))).(p - Proj(i,m).x) = reproj(i,z).r1
       by PDIFF_1:def 6;
   p - Proj(i,m).x = pm - Pr by REAL_NS1:5;
   then p - Proj(i,m).x = <* p1 - proj(i,m).y *> by A1,A4,RVSUM_1:29;
   hence (reproj(i,x)).p - x = reproj(i,0.(REAL-NS m)).(p - Proj(i,m).x)
     by A2,A3,A5,FINSEQ_1:76;
    x - (reproj(i,x)).p = y - rm by A1,REAL_NS1:5;
then A6:x - (reproj(i,x)).p = reproj(i,0*m).(proj(i,m).y - p1) by Th6;
   consider r2 be Element of REAL, z be Element of REAL m such that
A7: Proj(i,m).x - p = <* r2 *> & z = 0.(REAL-NS m)
    & (reproj(i,0.(REAL-NS m))).(Proj(i,m).x - p) = reproj(i,z).r2
       by PDIFF_1:def 6;
    Proj(i,m).x - p = Pr - pm by REAL_NS1:5;
   then Proj(i,m).x - p = <* proj(i,m).y - p1 *> by A1,A4,RVSUM_1:29;
   hence x - (reproj(i,x)).p = reproj(i,0.(REAL-NS m)).(Proj(i,m).x - p)
     by A3,A7,A6,FINSEQ_1:76;
end;
