reserve G for IncProjStr;
reserve a,a1,a2,b,b1,b2,c,d,p,q,r for POINT of G;
reserve A,B,C,D,M,N,P,Q,R for LINE of G;
reserve G for IncProjectivePlane;
reserve a,q for POINT of G;
reserve A,B for LINE of G;
reserve G for IncProjSp;
reserve a,b,c,d for POINT of G;
reserve P for LINE of G;

theorem Th26:
  a,b,c is_a_triangle implies a,b,c are_mutually_distinct
proof
  assume that
A1: a,b,c is_a_triangle and
A2: not a,b,c are_mutually_distinct;
  now
    per cases by A2,ZFMISC_1:def 5;
    case
A3:   a=b;
      ex P st b on P & c on P by INCPROJ:def 5;
      hence contradiction by A1,A3,Th5;
    end;
    case
A4:   b=c;
      ex P st a on P & b on P by INCPROJ:def 5;
      hence contradiction by A1,A4,Th5;
    end;
    case
A5:   c =a;
      ex P st b on P & c on P by INCPROJ:def 5;
      hence contradiction by A1,A5,Th5;
    end;
  end;
  hence thesis;
end;
