reserve m,k,j,j1,i,i1,i2,n for Nat,
  r,s for Real,
  C for compact non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  p for Point of TOP-REAL 2;

theorem Th17:
  for G being Y_increasing-in-line non empty-yielding Matrix of
  TOP-REAL 2 holds width G <= card(proj2.:Values G)
proof
  let G be Y_increasing-in-line non empty-yielding Matrix of TOP-REAL 2;
  0 <> len G by MATRIX_0:def 10;
  then 1 <= len G by NAT_1:14;
  then
A1: 1 in dom G by FINSEQ_3:25;
  then reconsider L = Y_axis(Line(G,1)) as increasing FinSequence of REAL by
GOBOARD1:def 6;
A2: card rng L= len L by FINSEQ_4:62
    .= len Line(G,1) by GOBOARD1:def 2
    .= width G by MATRIX_0:def 7;
A3: rng L = rng(proj2*Line(G,1)) by Th11
    .= proj2.:rng Line(G,1) by RELAT_1:127;
  rng Line(G,1) c= Values G by A1,Th13;
  hence thesis by A3,A2,NAT_1:43,RELAT_1:123;
end;
