reserve F for Field,
  a,b,c,d,e,f,g,h for Element of F;
reserve x,y for Element of [:the carrier of F,the carrier of F,the carrier of
  F:];
reserve F for Field;
reserve PS for non empty ParStr;
reserve x for set,
  a,b,c,d,e,f,g,h,i,j,k,l for Element of [:the carrier of F,
  the carrier of F,the carrier of F:];
reserve a,b,c,d,p,q,r,s for Element of MPS(F);
reserve PS for ParSp,
  a,b,c,d,p,q,r,s for Element of PS;

theorem
  (ex a,b st a<>b & for c holds a,b '||' a,c) implies for p,q,r,s holds
  p,q '||' r,s
proof
  assume ex a,b st a<>b & for c holds a,b '||' a,c;
  then consider a,b such that
A1: a<>b and
A2: for c holds a,b '||' a,c;
  let p,q,r,s;
  a,b '||' a,r & a,b '||' a,s by A2;
  then
A3: a,b '||' r,s by Th35;
  a,b '||' a,p & a,b '||' a,q by A2;
  then a,b '||' p,q by Th35;
  hence thesis by A1,A3,Def11;
end;
