reserve F for Field;
reserve a,b,c,d,p,q,r for Element of MPS(F);
reserve e,f,g,h,i,j,k,l,m,n,o,w for Element of [:the carrier of F,the carrier
  of F,the carrier of F:];
reserve K,L,M,N,R,S for Element of F;

theorem Th2:
  a,b '||' c,d iff ex e,f,g,h st [[a,b],[c,d]] = [[e,f],[g,h]] &
 ((ex K st K*(e`1_3-f`1_3) = g`1_3-h`1_3 & K*(e`2_3-f`2_3) = g`2_3-h`2_3 &
  K*(e`3_3-f`3_3) = g`3_3-h`3_3) or e`1_3-f
  `1_3 = 0.F & e`2_3-f`2_3 = 0.F & e`3_3-f`3_3 = 0.F )
proof
A1: a,b '||' c,d implies ex e,f,g,h st [[a,b],[c,d]] = [[e,f],[g,h]] &
 ((ex K st K*(e`1_3-f`1_3) = g`1_3-h`1_3 & K*(e`2_3-f`2_3) = g`2_3-h`2_3 &
  K*(e`3_3-f`3_3) = g`3_3-h`3_3) or e`1_3-f`1_3
  = 0.F & e`2_3-f`2_3 = 0.F & e`3_3-f`3_3 = 0.F )
  proof
    assume a,b '||' c,d;
    then consider e,f,g,h such that
A2: [[a,b],[c,d]] = [[e,f],[g,h]] and
A3: (e`1_3-f`1_3)*(g`2_3-h`2_3) - (g`1_3-h`1_3)*(e`2_3-f`2_3)
 = 0.F & (e`1_3-f`1_3)*(g`3_3-h
`3_3) - (g `1_3 -h`1_3)*(e`3_3-f`3_3) = 0.F &
  (e`2_3-f`2_3)*(g`3_3-h`3_3) - (g`2_3-h`2_3)*(e`3_3-f`3_3) =
    0.F by PARSP_1:12;
    (ex K st K*(e`1_3-f`1_3) = g`1_3-h`1_3 &
  K*(e`2_3-f`2_3) = g`2_3-h`2_3 & K*(e`3_3-f`3_3) =
    g`3_3-h`3_3) or e`1_3-f`1_3 = 0.F & e`2_3-f`2_3 = 0.F & e`3_3-f`3_3 = 0.F
  by A3,Lm3;
    hence thesis by A2;
  end;
  (ex e,f,g,h st [[a,b],[c,d]] = [[e,f],[g,h]] &
   ((ex K st K*(e`1_3-f`1_3) = g`1_3-h`1_3 &
K*(e`2_3-f`2_3) = g`2_3-h`2_3 &
   K*(e`3_3-f`3_3) = g`3_3-h`3_3) or (e`1_3-f`1_3 = 0.F & e`2_3-f`2_3 = 0.
  F & e`3_3-f`3_3 = 0.F))) implies a,b '||' c,d
  proof
    given e,f,g,h such that
A4: [[a,b],[c,d]] = [[e,f],[g,h]] and
A5: (ex K st K*(e`1_3-f`1_3) = g`1_3-h`1_3 &
      K*(e`2_3-f`2_3) = g`2_3-h`2_3 & K*(e`3_3-f
    `3_3 ) = g`3_3-h`3_3) or e`1_3-f`1_3 = 0.F &
         e`2_3-f`2_3 = 0.F & e`3_3-f`3_3 = 0.F;
A6: (e`2_3-f`2_3)*(g`3_3-h`3_3) - (g`2_3-h`2_3)*(e`3_3-f`3_3) = 0.F by A5,Lm3;
    (e`1_3-f`1_3)*(g`2_3-h`2_3) - (g`1_3-h`1_3)*(e`2_3-f`2_3) = 0.F &
   (e`1_3-f`1_3)*(g`3_3-h`3_3)
    - (g`1_3 -h`1_3)*(e`3_3-f`3_3) = 0.F by A5,Lm3;
    hence thesis by A4,A6,PARSP_1:12;
  end;
  hence thesis by A1;
end;
