reserve n for non zero Element of NAT;
reserve a,b,r,t for Real;

theorem Th28:
for n,i be Nat, f be PartFunc of REAL,REAL n,
 A be Subset of REAL holds proj(i,n)*(f|A) = (proj(i,n)*f)|A
proof
  let n,i be Nat, f be PartFunc of REAL,REAL n, A be Subset of REAL;
A1: dom (proj(i,n))=REAL n by FUNCT_2:def 1;
  then
A2: rng f c= dom(proj(i,n));
A3: rng (f|A) c= dom (proj(i,n)) by A1;
A4: now
    let c be object;
    assume
A5: c in dom ((proj(i,n)*f)|A); then
A6: c in dom (proj(i,n)*f) /\ A by RELAT_1:61;
A7: c in dom (proj(i,n)*f) by A6,XBOOLE_0:def 4;
    then c in dom f by A2,RELAT_1:27;
    then c in dom f /\ A by A5,XBOOLE_0:def 4;then
A8: c in dom (f|A) by RELAT_1:61;
    then c in dom (proj(i,n)*(f|A)) by A3,RELAT_1:27;
    then (proj(i,n)*(f|A)).c = (proj(i,n)).((f|A).c) by FUNCT_1:12
      .= (proj(i,n)).(f.c) by A8,FUNCT_1:47
      .= (proj(i,n)*f).c by A7,FUNCT_1:12;
    hence ((proj(i,n)*f)|A).c = (proj(i,n)*(f|A)).c by A5,FUNCT_1:47;
  end;
  dom ((proj(i,n)*f)|A) = dom (proj(i,n)*f) /\ A by RELAT_1:61
    .= dom f /\ A by A2,RELAT_1:27
    .= dom (f|A) by RELAT_1:61
    .= dom (proj(i,n)*(f|A)) by A3,RELAT_1:27;
  hence thesis by A4,FUNCT_1:2;
end;
