theorem Satz7p22:
  between a1,c,a2 & between b1,c,b2 & c,a1 equiv c,b1 &
  c,a2 equiv c,b2 & Middle a1,m1,b1 & Middle a2,m2,b2 implies between m1,c,m2
  proof
    assume that
A1: between a1,c,a2 and
A2: between b1,c,b2 and
A3: c,a1 equiv c,b1 and
A4: c,a2 equiv c,b2 and
A5: Middle a1,m1,b1 and
A6: Middle a2,m2,b2;
    per cases by Satz5p10;
    suppose c,a1 <= c,a2;
      hence thesis by A1,A2,A3,A4,A5,A6,Satz7p22part1;
    end;
    suppose c,a2 <= c,a1;
      hence thesis by A1,A2,A3,A4,A5,A6,Satz7p22part2;
    end;
  end;
