
theorem Th7:
  for F being Function, i1,i2,xi1 being set, Ai2 being Subset of F.
  i2 st product F <> {} & xi1 in F.i1 & i1 in dom F & i2 in dom F & Ai2<>F.i2
  holds proj(F,i1)"({xi1}) c= proj(F,i2)"Ai2 iff i1 = i2 & xi1 in Ai2
proof
  let F be Function, i1,i2,xi1 be set, Ai2 be Subset of F.i2;
  assume that
A1: product F <> {} and
A2: xi1 in F.i1 and
A3: i1 in dom F and
A4: i2 in dom F and
A5: Ai2<>F.i2;
  set f9 = the Element of product F;
  consider f being Function such that
A6: f9 = f and
A7: dom f = dom F and
  for x being object st x in dom F holds f.x in F.x by A1,CARD_3:def 5;
  not F.i2 c= Ai2 by A5;
  then consider xi2 being object such that
A8: xi2 in F.i2 and
A9: not xi2 in Ai2;
  reconsider xi2 as Element of F.i2 by A8;
A10: (f+*(i2,xi2)).i2 = xi2 by A4,A7,FUNCT_7:31;
  thus proj(F,i1)"({xi1}) c= proj(F,i2)"Ai2 implies i1 = i2 & xi1 in Ai2
  proof
    assume
A11: proj(F,i1)"({xi1}) c= proj(F,i2)"Ai2;
    thus
A12: i1 = i2
    proof
      assume
A13:  i1<>i2;
      f+*(i2,xi2) in product F & (f+*(i2,xi2))+*(i1,xi1) in proj(F,i1)"({
      xi1}) by A1,A2,A3,A8,A6,Th2,Th5;
      then f+*(i2,xi2) in proj(F,i2)"Ai2 by A2,A11,A13,Th6;
      then f+*(i2,xi2) in dom proj(F,i2) & proj(F,i2).(f+*(i2,xi2)) in Ai2 by
FUNCT_1:def 7;
      hence contradiction by A9,A10,CARD_3:def 16;
    end;
    xi1 in rng proj(F,i1) by A1,A2,A3,Th3;
    then {xi1} c= rng proj(F,i1) by ZFMISC_1:31;
    then {xi1} c= Ai2 by A11,A12,FUNCT_1:88;
    hence thesis by ZFMISC_1:31;
  end;
  assume that
A14: i1 = i2 and
A15: xi1 in Ai2;
  {xi1} c= Ai2 by A15,ZFMISC_1:31;
  hence thesis by A14,RELAT_1:143;
end;
