reserve X for non empty set,
  F for with_the_same_dom Functional_Sequence of X, ExtREAL,
  seq,seq1,seq2 for ExtREAL_sequence,
  x for Element of X,
  a,r for R_eal,
  n,m,k for Nat;
reserve S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S;

theorem Th9:
  for Y being non empty Subset of ExtREAL, r be R_eal st r in REAL
  holds inf({r} + Y) = inf {r} + inf Y
proof
  let Y be non empty Subset of ExtREAL;
  let r be R_eal;
  set X = {r};
  assume
A1: r in REAL;
  set W = X+Y;
  set Z = -X;
A2: -r <> -infty by A1,XXREAL_3:23;
A3: -r <> +infty by A1,XXREAL_3:23;
  now
    let z be object;
    assume z in Z + W;
    then consider x,y be R_eal such that
A4: z = x + y and
A5: x in Z and
A6: y in W;
    consider x1,y1 be R_eal such that
A7: y = x1 + y1 and
A8: x1 in X and
A9: y1 in Y by A6;
    -x in -Z by A5;
    then -x in X;
    then -x = r by TARSKI:def 1;
    then z = -r + (r + y1) by A4,A8,A7,TARSKI:def 1;
    then z = (-r + r) + y1 by A1,A3,A2,XXREAL_3:29;
    then z = 0. + y1 by XXREAL_3:7;
    hence z in Y by A9,XXREAL_3:4;
  end;
  then
A10: Z + W c= Y by TARSKI:def 3;
A11: r in X by TARSKI:def 1;
  now
    let z be object;
    assume
A12: z in Y;
    then reconsider y = z as Element of ExtREAL;
    r + y - r = (-r + r) + y by A1,A3,A2,XXREAL_3:29;
    then r + y - r = 0. + y by XXREAL_3:7;
    then
A13: y = r + y - r by XXREAL_3:4;
A14: -r in Z by A11;
    r + y in W by A11,A12;
    hence z in Z + W by A14,A13;
  end;
  then Y c= Z + W by TARSKI:def 3;
  then Y = Z + W by A10,XBOOLE_0:def 10;
  then inf Y >= inf W + inf Z by SUPINF_2:9;
  then inf Y >= inf(X+Y) + -sup X by SUPINF_2:14;
  then inf Y >= inf(X+Y) -r by XXREAL_2:11;
  then r + inf Y >= inf(X+Y) by A1,XXREAL_3:52;
  then
A15: inf X + inf Y >= inf(X+Y) by XXREAL_2:13;
  inf(X + Y) >= inf X + inf Y by SUPINF_2:9;
  hence thesis by A15,XXREAL_0:1;
end;
