reserve x,y,y1,y2,z,e,s for set;
reserve alpha,beta,gamma for Ordinal;
reserve n,m,k for Nat;
reserve g,g0,g1,g2,gO,gL,gR,gLL,gLR,gRL,gRR for ConwayGame;

theorem Th5:
  g1 = g2 iff
    (the_LeftOptions_of g1 = the_LeftOptions_of g2
      & the_RightOptions_of g1 = the_RightOptions_of g2)
proof
  thus g1 = g2 implies (the_LeftOptions_of g1 = the_LeftOptions_of g2
                        & the_RightOptions_of g1 = the_RightOptions_of g2);
  reconsider w1 = g1 as strict left-right by Th4;
  reconsider w2 = g2 as strict left-right by Th4;
  assume
A1: the_LeftOptions_of g1 = the_LeftOptions_of g2
      & the_RightOptions_of g1 = the_RightOptions_of g2;
  the LeftOptions of w1 = the_LeftOptions_of g1
    & the LeftOptions of w2 = the_LeftOptions_of g2
    & the RightOptions of w1 = the_RightOptions_of g1
    & the RightOptions of w2 = the_RightOptions_of g2
    by Def6,Def7;
  hence g1 = g2 by A1;
end;
