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 Th51:
  f in doms <*g*> iff len f = 1 & f.1 in dom g
proof
  set G=<*g*>;
A1: len G=1 & G.1=g by FINSEQ_1:40;
  hereby assume
A2:    f in doms G;
     then len f = 1 by A1,Th47;
     then 1 in dom f by FINSEQ_3:25;
     hence len f = 1 & f.1 in dom g by A1,A2,Th47;
  end;
  assume
A3: len f = 1 & f.1 in dom g;
  now let i such that
A4:    i in dom f;
     1<= i & i <= 1 by A3,A4,FINSEQ_3:25;
     then i=1 by XXREAL_0:1;
    hence f.i in dom (G.i) by A3;
  end;
  hence thesis by A3,A1,Th47;
end;
