reserve m,n for non zero Element of NAT;
reserve i,j,k for Element of NAT;
reserve Z for set;

theorem Th12:
for x be Element of REAL j holds x = reproj(i,x).(proj(i,j).x)
proof
   let x be Element of REAL j;
   set q=reproj(i,x).(proj(i,j).x);
A1:dom q = Seg j & dom x = Seg j by FINSEQ_1:89;
A2:len x = j by A1,FINSEQ_1:def 3;
   for k be Nat st k in dom x holds x.k = q.k
   proof
    let k be Nat;
    assume A3: k in dom x; then
A4: 1 <= k & k <= j by A1,FINSEQ_1:1;
    q = Replace(x,i,proj(i,j).x) by PDIFF_1:def 5; then
A5: q.k = Replace(x,i,proj(i,j).x)/.k by A3,A1,PARTFUN1:def 6;
    per cases;
    suppose A6: k = i; then
     q.k = proj(i,j).x by A2,A4,A5,FINSEQ_7:8;
     hence x.k = q.k by A6,PDIFF_1:def 1;
    end;
    suppose k <> i; then
     q.k = x/.k by A2,A4,A5,FINSEQ_7:10;
     hence x.k = q.k by A3,PARTFUN1:def 6;
    end;
   end;
   hence  x = reproj(i,x).(proj(i,j).x) by A1,FINSEQ_1:13;
end;
