reserve
  a for natural Number,
  k,l,m,n,k1,b,c,i for Nat,
  x,y,z,y1,y2 for object,
  X,Y for set,
  f,g for Function;

theorem Th5:
  for a,b being natural Number holds a <= b iff Seg a c= Seg b
proof
  let a,b be natural Number;
  thus a <= b implies Seg a c= Seg b
  proof
    assume
A1: a <= b;
    let x be object;
    assume x in Seg a;
    then consider c being Nat such that
A3: x = c & 1 <= c & c <= a;
    c <= b by A1,A3,XXREAL_0:2;
    hence thesis by A3;
  end;
  assume
A4: Seg a c= Seg b;
  now
    assume a <> 0;
    then a in Seg a by Th3;
    hence thesis by A4,Th1;
  end;
  hence thesis;
end;
