reserve S for non empty satisfying_CongruenceIdentity
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              satisfying_Pasch
              TarskiGeometryStruct;
reserve a,b for POINT of S;
reserve A for Subset of S;
reserve S for non empty satisfying_Tarski-model
              TarskiGeometryStruct;
reserve a,b,c,m,r,s for POINT of S;
reserve A for Subset of S;

theorem Th7:
  between a,A,c & m in A & Middle a,m,c & r in A implies
  (for b st r out a,b holds between b,A,c)
  proof
    assume that
A1: between a,A,c and
A2: m in A and
A3: Middle a,m,c and
A4: r in A;
    let b;
    assume
A5: r out a,b;
    then per cases;
    suppose between r,b,a;
      hence thesis by A1,A2,A3,A4,A5,Th6;
    end;
    suppose
A6:   between r,a,b;
      set b9 = reflection(m,b),r9 = reflection(m,r);
A7:   Middle b,m,b9 & Middle r,m,r9 by GTARSKI3:def 13;
A8:   between reflection(m,r),reflection(m,a),reflection(m,b)
        by A6,GTARSKI3:106;
      then
A9:   between reflection(m,r),c,reflection(m,b) by A3,GTARSKI3:def 13;
A10:     A is_line & not b in A by A1,A4,A6,Th4;
         G1: not b9 in A
          proof
            assume
A11:        b9 in A;
            b9 <> m
            proof
              assume b9 = m;
              then Middle b,m,m by GTARSKI3:100;
              hence contradiction by A10,A2,GTARSKI1:def 7;
            end;
            hence contradiction by A10,A11,A7,A2,Th5,GTARSKI3:14;
          end;
        Middle b9,m,b by GTARSKI3:def 13,GTARSKI3:96;
        then U1: between b9,A,b by A2,G1,A1,A4,A6,Th4;
        U2: Middle b9,m,b by GTARSKI3:96,def 13;
        U3: r9 in A by A4,GTARSKI3:104,A1,A2,A7,Th5;
        then U4: r9 out b9,c by A9,A1,GTARSKI1:def 10;
         between r9,c,b9 by A8,A3,GTARSKI3:def 13;
      then between c,A,b by U1,U2,U3,U4,A2,Th6;
      hence thesis by GTARSKI3:14;
    end;
  end;
