let G be Group; :: thesis:
let a be Element of G; :: thesis:
assume A1: ( a, 1_ G are_conjugated or 1_ G,a are_conjugated ) ; :: thesis:

now :: thesis:

per cases by A1;

suppose a, 1_ G are_conjugated ; :: thesis:

then ex g being Element of G st 1_ G = a |^ g by Th74;
hence a = 1_ G by Th18; :: thesis:

end;

suppose 1_ G,a are_conjugated ; :: thesis:

then ex g being Element of G st 1_ G = a |^ g ;
hence a = 1_ G by Th18; :: thesis:

end;

end;

end;

hence a = 1_ G ; :: thesis: