reserve x,y,y1,y2 for set;
reserve G for Group;
reserve a,b,c,d,g,h for Element of G;
reserve A,B,C,D for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve n for Nat;
reserve i for Integer;

theorem Th78:
  a,1_G are_conjugated or 1_G,a are_conjugated implies a = 1_G
proof
  assume
A1: a,1_G are_conjugated or 1_G,a are_conjugated;
  now
    per cases by A1;
    suppose
      a,1_G are_conjugated;
      then ex g st 1_G = a |^ g by Th74;
      hence thesis by Th18;
    end;
    suppose
      1_G,a are_conjugated;
      then ex g st 1_G = a |^ g;
      hence thesis by Th18;
    end;
  end;
  hence thesis;
end;
