reserve r,lambda for Real,
  i,j,n for Nat;
reserve p,p1,p2,q1,q2 for Point of TOP-REAL 2,
  P, P1 for Subset of TOP-REAL 2;
reserve T for TopSpace;
reserve f,f1,f2,h for FinSequence of TOP-REAL 2;

theorem Th37:
  west_halfline p = { |[ r,p`2 ]| where r is Real: r <= p`1 }
proof
  set A = {|[ r,p`2 ]| where r is Real: r <= p`1};
  hereby
    let x be object;
    assume
A1: x in west_halfline p;
    then reconsider q = x as Point of TOP-REAL 2;
A2: q`2 = p`2 by A1,Def13;
A3: q = |[q`1, q`2]| by EUCLID:53;
    q`1 <= p`1 by A1,Def13;
    hence x in A by A2,A3;
  end;
  let x be object;
  assume x in A;
  then consider r being Real such that
A4: x = |[r,p`2]| and
A5: r <= p`1;
  reconsider q = x as Point of TOP-REAL 2 by A4;
A6: q`2 = p`2 by A4,EUCLID:52;
  q`1 = r by A4,EUCLID:52;
  hence thesis by A5,A6,Def13;
end;
