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 Th14:
  for G being X_increasing-in-column non empty-yielding Matrix of
  TOP-REAL 2 holds len G <= card(proj1.:Values G)
proof
  let G be X_increasing-in-column non empty-yielding Matrix of TOP-REAL 2;
  0 <> width G by MATRIX_0:def 10;
  then 1 <= width G by NAT_1:14;
  then
A1: 1 in Seg width G by FINSEQ_1:1;
  then reconsider L = X_axis(Col(G,1)) as increasing FinSequence of REAL by
GOBOARD1:def 7;
A2: card rng L= len L by FINSEQ_4:62
    .= len Col(G,1) by GOBOARD1:def 1
    .= len G by MATRIX_0:def 8;
A3: rng L = rng(proj1*Col(G,1)) by Th10
    .= proj1.:rng Col(G,1) by RELAT_1:127;
  rng Col(G,1) c= Values G by A1,Th12;
  hence thesis by A3,A2,NAT_1:43,RELAT_1:123;
end;
