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 Th32:
  for G being Go-board holds G = GoB(SgmX(RealOrd, proj1.:Values G
  ), SgmX(RealOrd,proj2.:Values G))
proof
  let G be Go-board;
  set v1 = SgmX(RealOrd, proj1.:Values G), v2 = SgmX(RealOrd,proj2.:Values G);
A1: width G = card(proj2.:Values G) by Th19
    .= len v2 by Th9;
A2: for n,m st [n,m] in Indices G holds G*(n,m) = |[v1.n,v2.m]|
  proof
    let n,m;
    assume
A3: [n,m] in Indices G;
    then
A4: 1 <= n & n <= len G by MATRIX_0:32;
A5: 1 <= m & m <= width G by A3,MATRIX_0:32;
    v1.n = G*(n,1)`1 by A4,Th28;
    then
A6: v1.n = G*(n,m)`1 by A4,A5,GOBOARD5:2;
    v2.m = G*(1,m)`2 by A5,Th29;
    then v2.m = G*(n,m)`2 by A4,A5,GOBOARD5:1;
    hence thesis by A6,EUCLID:53;
  end;
  len G = card(proj1.:Values G) by Th16
    .= len v1 by Th9;
  hence thesis by A1,A2,GOBOARD2:def 1;
end;
