reserve V for RealLinearSpace;
reserve u,u1,u2,v,v1,v2,w,w1,y for VECTOR of V;
reserve a,a1,a2,b,b1,b2,c1,c2 for Real;
reserve x,z for set;
reserve a,b,c,d,a1,a2,b1,c1,d1,d2,p,q for Element of DTrSpace(V,w,y);
reserve OTS for OrdTrapSpace;
reserve a,b,c,d for Element of OTS;
reserve a9,b9,c9,d9,a19,b19,d19 for Element of Lambda(OTS);

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;
