reserve X,x,y,z for set;
reserve n,m,k,k9,d9 for Nat;

theorem Th1:
  for x,y being Real st x < y
ex z being Element of REAL st x < z & z < y
proof
  let x,y be Real;
  assume x < y;
  then consider z being Real such that
A1: x < z and
A2: z < y by XREAL_1:5;
  reconsider z as Element of REAL by XREAL_0:def 1;
  take z;
  thus thesis by A1,A2;
end;
