theorem Th103:
  B is having_a_unity associative commutative having_an_inverseOp
implies
    (the_inverseOp_wrt B).(B.(d1,d2)) =
         B.((the_inverseOp_wrt B).d1,(the_inverseOp_wrt B).d2)
proof
  set I=the_inverseOp_wrt B;
  assume
A1: B is having_a_unity associative commutative having_an_inverseOp;
  B. (B.(d1,d2), B.(I.d1,I.d2)) = B. (B.(d2,d1), B.(I.d1,I.d2))
    by A1,BINOP_1:def 2
  .= B.(B.(B.(d2,d1),I.d1),I.d2) by A1,BINOP_1:def 3
  .= B.(B.(d2,B.(d1,I.d1)),I.d2) by A1,BINOP_1:def 3
  .= B.(B.(d2,the_unity_wrt B),I.d2) by A1,FINSEQOP:59
  .= B.(d2,I.d2) by A1,SETWISEO:15
  .= the_unity_wrt B by A1,FINSEQOP:59;
  hence thesis by A1,FINSEQOP:60;
end;
