reserve F for Field;
reserve S for SymSp of F;
reserve a,b,c,d,a9,b9,p,q,r,s,x,y,z for Element of S;
reserve k,l for Element of F;

theorem Th10:
  1_F+1_F<>0.F & a<>0.S & b<>0.S & c <>0.S implies ex p st not p
  _|_ a & not p _|_ b & not p _|_ c
proof
  assume that
A1: 1_F+1_F<>0.F and
A2: a<>0.S and
A3: b<>0.S and
A4: c <>0.S;
  consider r such that
A5: not r _|_ a and
A6: not r _|_ b by A2,A3,Th9;
  consider s such that
A7: not s _|_ a and
A8: not s _|_ c by A2,A4,Th9;
  now
    assume that
A9: r _|_ c and
A10: s _|_ b;
A11: now
      (1_F+1_F)*r _|_ c by A9,Def1;
      then
A12:  not (1_F+1_F)*r+s _|_ c by A8,Th4;
      not (1_F+1_F)*r _|_ b by A1,A6,Th5;
      then
A13:  not (1_F+1_F)*r+s _|_ b by A10,Th4;
      assume not (1_F+1_F)*r+s _|_ a;
      hence thesis by A13,A12;
    end;
    now
      assume
A14:  not r+s _|_ a;
      ( not r+s _|_ b)& not r+s _|_ c by A6,A8,A9,A10,Th4;
      hence thesis by A14;
    end;
    hence thesis by A5,A11,Th7;
  end;
  hence thesis by A5,A6,A7,A8;
end;
