reserve a for set;
reserve L for lower-bounded sup-Semilattice;
reserve x for Element of L;
reserve L for complete LATTICE;
reserve AR for Relation of L;
reserve x, y, z for Element of L;

theorem
  ( for x holds ( ex y st y is_maximal_wrt (AR-below x), AR )
  implies [x,x] in AR ) implies AR is satisfying_SI
proof
  assume
A1: for x holds ( ex y st y is_maximal_wrt (AR-below x), AR )
  implies [x,x] in AR;
  now
    let z,x;
    assume that
A2: [z,x] in AR and
A3: z <> x;
A4: z in AR-below x by A2;
    per cases;
    suppose [x,x] in AR;
      hence ex y st [z,y] in AR & [y,x] in AR & z <> y by A2,A3;
    end;
    suppose not [x,x] in AR;
      then not z is_maximal_wrt (AR-below x), AR by A1;
      then consider y being set such that
A5:   y in AR-below x and
A6:   y <> z and
A7:   [z,y] in AR by A4;
      [y,x] in AR by A5,Th13;
      hence ex y st [z,y] in AR & [y,x] in AR & z <> y by A5,A6,A7;
    end;
  end;
  hence thesis;
end;
