reserve i1 for Element of INT;
reserve j1,j2,j3 for Integer;
reserve p,s,k,n for Nat;
reserve x,y,xp,yp for set;
reserve G for Group;
reserve a,b for Element of G;
reserve F for FinSequence of G;
reserve I for FinSequence of INT;

theorem Th17:
  G is cyclic Group iff ex a being Element of G st
    for b being Element of G ex j1 st b=a|^j1
proof
  thus G is cyclic Group implies ex a being Element of G st for b being
  Element of G ex j1 st b=a|^j1
  proof
    assume G is cyclic Group;
    then consider a being Element of G such that
A1: the multMagma of G = gr {a} by Def7;
    take a;
    for b being Element of G ex j1 st b=a|^j1 by A1,Th5,STRUCT_0:def 5;
    hence thesis;
  end;
  given a being Element of G such that
A2: for b being Element of G ex j1 st b = a|^j1;
  for b being Element of G holds b in gr {a}
  proof
    let b be Element of G;
    ex j1 st b=a|^j1 by A2;
    hence thesis by Th5;
  end;
  then the multMagma of G = gr {a} by GROUP_2:62;
  hence thesis by Def7;
end;
