theorem Th114:
  a * H = b * H iff b" * a in H
proof
  thus a * H = b * H implies b" * a in H
  proof
    assume
A1: a * H = b * H;
    b" * a * H = b" * (a * H) by Th32
      .= b" * b * H by A1,Th32
      .= 1_G * H by GROUP_1:def 5
      .= carr(H) by Th37;
    hence thesis by Th113;
  end;
  assume
A2: b" * a in H;
  thus a * H = 1_G * (a * H) by Th37
    .= 1_G * a * H by Th32
    .= b * b" * a * H by GROUP_1:def 5
    .= b * (b" * a) * H by GROUP_1:def 3
    .= b * ((b" * a) * H) by Th32
    .= b * H by A2,Th113;
end;
