
theorem Th32:
  for A being real-membered set, x being Real,
      y being R_eal st x = y holds inf(x ++ A) = y + inf A
proof
  let A be real-membered set, x being Real, y being R_eal such that
A1: x = y;
A2: for z being LowerBound of x ++ A holds z <= y + inf A
  proof
    let z be LowerBound of x ++ A;
    reconsider zz = z as R_eal by XXREAL_0:def 1;
    zz - y is LowerBound of A
    proof
      let u be ExtReal;
      reconsider uu = u as R_eal by XXREAL_0:def 1;
      assume
A3:   u in A;
      then reconsider u1 = u as Real;
      y + uu = x + u1 by A1,XXREAL_3:def 2;
      then y + uu in x ++ A by A3,Lm1;
      then z <= y + uu by XXREAL_2:def 2;
      hence thesis by A1,XXREAL_3:42;
    end;
    then zz - y <= inf A by XXREAL_2:def 4;
    hence thesis by A1,XXREAL_3:41;
  end;
  y + inf A is LowerBound of x ++ A
  proof
    let z be ExtReal;
    assume z in x ++ A;
    then consider a being Real such that
A4: a in A and
A5: z = x + a by Lm1;
    reconsider b = a as R_eal by XXREAL_0:def 1;
A6: inf A <= a by A4,XXREAL_2:3;
    ex a,c being Complex st y = a & b = c & y + b = a + c by A1,
XXREAL_3:def 2;
    hence thesis by A1,A5,A6,XXREAL_3:36;
  end;
  hence thesis by A2,XXREAL_2:def 4;
end;
