reserve i,j,n,k,m for Nat,
     a,b,x,y,z for object,
     F,G for FinSequence-yielding FinSequence,
     f,g,p,q for FinSequence,
     X,Y for set,
     D for non empty set;

theorem Th45:
  doms F is non empty iff F is non-empty
proof
  thus doms F is non empty implies F is non-empty
  proof
    assume doms F is non empty;
    then consider x be object such that
A1:   x in doms F;
    consider p be FinSequence such that
A2:   p=x & len p = len F &
        for i st i in dom p holds p.i in dom (F.i) by A1,Def8;
    let d be object such that
A3:   d in dom F;
    dom p=dom F by A2,FINSEQ_3:29;
    then p.d in dom (F.d) by A2,A3;
    hence thesis;
  end;
  set L = len F |-> 1;
  assume
A4: F is non-empty;
A5: len L=len F;
  then
A6: dom L=dom F by FINSEQ_3:29;
  for i st i in dom L holds L.i in dom (F.i)
  proof
    let i such that
A7:   i in dom L;
A8:   L.i = 1 by A7,FINSEQ_2:57;
    F.i is non empty by A4,A7,A6;
    then len (F.i) >=1 by NAT_1:14;
    hence thesis by A8,FINSEQ_3:25;
  end;
  hence thesis by Def8,A5;
end;
