theorem Th39:
for m be non zero Nat, x,y be Element of REAL m,
    i be Nat,
    xi be Real st
 1 <=i & i <= m & y=reproj(i,x).xi holds proj(i,m).y = xi
proof
   let m be non zero Nat,
       x,y be Element of REAL m,
       i be Nat,
       xi be Real;
    reconsider xx=xi as Element of REAL by XREAL_0:def 1;
   assume A1: 1 <= i & i <= m & y = reproj(i,x).xi;
then A2:y = Replace(x,i,xx) by PDIFF_1:def 5;
A3:len x = m & len y = m by CARD_1:def 7;
then A4:i in dom y by A1,FINSEQ_3:25;
    y/.i = xi by A1,A2,A3,FINSEQ_7:8;
   then y.i = xi by A4,PARTFUN1:def 6;
   hence proj(i,m).y = xi by PDIFF_1:def 1;
end;
