reserve e,u for set;
reserve X, Y for non empty TopSpace;

theorem Th40:
  the carrier of I[01] = [.0,1.]
proof
  reconsider P = [.0,1.] as Subset of TopSpaceMetr RealSpace by Lm3,
METRIC_1:def 13;
A1: I[01] = (TopSpaceMetr RealSpace)|P by Def13;
  thus the carrier of I[01] = [#] I[01] .= [.0,1.] by A1,PRE_TOPC:def 5;
end;
