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 Th55:
  doms <*f*> = { <*i*> where i is Element of NAT:i in dom f}
proof
  thus doms <*f*> c= { <*i*> where i is Element of NAT:i in dom f}
  proof
    let y;
    assume
A1:   y in doms <*f*>;
    then reconsider y as FinSequence;
A2:   y.1 in dom f by A1,Th51;
    y = <*y.1*> by A1,Th51,FINSEQ_1:40;
    hence thesis by A2;
  end;
  let y;
  assume y in { <*i*> where i is Element of NAT:i in dom f};
  then consider i be Element of NAT  such that
A3: y = <*i*> & i in dom f;
  len <*i*> = 1 & <*i*>.1=i by FINSEQ_1:40;
  hence thesis by A3,Th51;
end;
