theorem
  dim (n-VectSp_over K) = n
proof
  set ONE=1.(K,n);
  len ONE=n by MATRIX_0:24;
  then
A1: dom ONE=Seg n by FINSEQ_1:def 3;
  then
A2: ONE.:Seg n=lines ONE by RELAT_1:113;
  the_rank_of ONE=n by Lm8;
  then ONE is without_repeated_line by Th105;
  then Seg n,ONE.:Seg n are_equipotent by A1,CARD_1:33;
  then card Seg n=card lines ONE by A2,CARD_1:5;
  then
A3: card lines ONE=n by FINSEQ_1:57;
  lines ONE is Basis of n-VectSp_over K by Lm8;
  hence thesis by A3,VECTSP_9:def 1;
end;
