reserve SAS for Semi_Affine_Space;
reserve a,a9,a1,a2,a3,a4,b,b9,c,c9,d,d9,d1,d2,o,p,p1,p2,q,r,r1,r2,s,x, y,t,z
  for Element of SAS;

theorem Th68:
  congr a,b,c,d implies congr a,c,b,d
proof
  assume
A1: congr a,b,c,d;
A2: now
    assume
A3: a=c;
    congr a,b,a,b by Th64;
    then b=d by A1,A3,Th62;
    hence thesis by A3;
  end;
A4: now
    assume that
A5: a<>b and
A6: a<>c and
A7: a,b,c are_collinear;
A8: a,b // a,c by A7;
    consider p,q such that
A9: parallelogram p,q,a,b and
A10: parallelogram p, q,c,d by A1,A5;
A11: a,p // a,p by Th1;
    ( not a,b,p are_collinear)& a<>p by A9,Th36,Th38;
    then consider r such that
A12: parallelogram a,c,p,r by A6,A8,A11,Th23,Th44;
A13: a,c // p,r by A12;
A14: p,q // c,d by A10;
    p<>q & p,q // a,b by A9,Th36;
    then
A15: a,b // c,d by A14,Def1;
    then a,c // b,d by A5,A7,Th32;
    then
A16: p,r // b,d by A6,A13,Def1;
    parallelogram p,r,a,c by A12,Th43;
    then
A17: p,a // r,c;
    p,a // q,b & p<>a by A9,Th36;
    then
A18: r,c // q,b by A17,Def1;
    p,c // q,d by A10;
    then
A19: q,d // p,c by Th6;
    p,q // a,b by A9;
    then
A20: a,b // p,q by Th6;
A21: a,c // p,r by A12;
    a,b // a,c by A7;
    then a,c // p,q by A5,A20,Def1;
    then p,q // p,r by A6,A21,Def1;
    then
A22: r,q // r,p by Th7;
    a,c,b are_collinear by A7,Th22;
    then
A23: not p,r,b are_collinear by A12,Th39;
A24: parallelogram p,r,a,c by A12,Th43;
    c,b // c,d by A5,A7,A15,Th33;
    then b,c // b,d by Th7;
    then p,b // r,d by A22,A18,A19,Def1;
    then parallelogram p,r,b,d by A23,A16;
    hence thesis by A24;
  end;
A25: now
    assume that
    a<>b and
    a<>c and
A26: not a,b,c are_collinear;
    parallelogram a,b,c,d by A1,A26,Th59;
    then parallelogram a,c,b,d by Th43;
    hence thesis by Th60;
  end;
  now
    assume
A27: a=b;
    then c =d by A1,Th54;
    hence thesis by A27,Th64;
  end;
  hence thesis by A2,A4,A25;
end;
