theorem Satz7p20:
  Collinear a,m,b & m,a equiv m,b implies a = b or Middle a,m,b
  proof
    assume that
A1: Collinear a,m,b and
A2: m,a equiv m,b;
    assume
A3: a <> b;
    per cases by A1;
    suppose between a,m,b;
      hence thesis by A2;
    end;
    suppose between m,b,a;
      then
A4:   between a,b,m by Satz3p2;
A5:   between b,b,m by Satz3p1,Satz3p2;
      m,a equiv b,m by A2,Satz2p5; then
A6:   a,m equiv b,m by Satz2p4;
      b,m equiv b,m by Satz2p1;
      hence thesis by A3,A4,A5,A6,Satz4p3,GTARSKI1:def 7;
    end;
    suppose
A7:   between b,a,m;
A8:   between a,a,m by Satz3p1,Satz3p2;
      m,a equiv b,m by A2,Satz2p5;
      then a,m equiv b,m by Satz2p4; then
A9:   b,m equiv a,m by Satz2p2;
      a,m equiv a,m by Satz2p1;
      hence thesis by A3,A7,A8,A9,Satz4p3,GTARSKI1:def 7;
    end;
  end;
