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 Th19:
for x,y be Element of REAL, z be Element of REAL m, i be Nat st
 1 <= i & i <= m & y = proj(i,m).z holds
   reproj(i,z).x - z = reproj(i,0*m).(x-y) &
   z - reproj(i,z).x = reproj(i,0*m).(y-x)
proof
   let x,y be Element of REAL,z be Element of REAL m,
       i be Nat;
    reconsider xy = x-y, yx = y-x as Element of REAL;
   assume 1 <= i & i <= m & y=proj(i,m).z;
then A1:Replace(z,i,x) - z = Replace(0*m,i,xy)
   & z - Replace(z,i,x) = Replace(0*m,i,yx) by Th12;
    Replace(z,i,x) = reproj(i,z).x by PDIFF_1:def 5;
   hence thesis by A1,PDIFF_1:def 5;
end;
