
theorem Th6:
  for F,f being Function, i1,i2,xi1 being set, Ai2 being Subset of
  F.i2 st xi1 in F.i1 & f in product F holds i1 <> i2 implies (f in proj(F,i2)"
  Ai2 iff f+*(i1,xi1) in proj(F,i2)"Ai2)
proof
  let F,f be Function, i1,i2,xi1 be set, Ai2 be Subset of F.i2;
  assume that
A1: xi1 in F.i1 and
A2: f in product F;
  assume
A3: i1 <> i2;
  thus f in proj(F,i2)"Ai2 implies f+*(i1,xi1) in proj(F,i2)"Ai2
  proof
A4: (f+*(i1,xi1))+*(i1,f.i1) = f +*(i1,f.i1) by FUNCT_7:34
      .= f by FUNCT_7:35;
    assume f in proj(F,i2)"Ai2;
    hence thesis by A1,A2,A3,A4,Lm1,Th2;
  end;
  assume f+*(i1,xi1) in proj(F,i2)"Ai2;
  hence thesis by A2,A3,Lm1;
end;
