theorem Th48:
  a=a9 & b=b9 & c =c9 & d=d9 implies (a9,b9 // c9,d9 iff a,b // c,
  d or a,b // d,c)
proof
A1: Lambda(OTS) = AffinStruct(#the carrier of OTS,lambda(the CONGR of OTS)#)
  by DIRAF:def 2;
  assume
A2: a=a9 & b=b9 & c =c9 & d=d9;
  hereby
    assume a9,b9 // c9,d9;
    then [[a9,b9],[c9,d9]] in lambda(the CONGR of OTS) by A1,ANALOAF:def 2;
    then
    [[a,b],[c,d]] in the CONGR of OTS or [[a,b],[d,c]] in the CONGR of OTS
    by A2,DIRAF:def 1;
    hence a,b // c,d or a,b // d,c by ANALOAF:def 2;
  end;
  assume a,b // c,d or a,b // d,c;
  then [[a,b],[c,d]] in the CONGR of OTS or [[a,b],[d,c]] in the CONGR of OTS
  by ANALOAF:def 2;
  then [[a,b],[c,d]] in the CONGR of Lambda(OTS) by A1,DIRAF:def 1;
  hence thesis by A2,ANALOAF:def 2;
end;
