reserve u,v,x,y,z,X,Y for set;
reserve r,s for Real;
reserve N for non empty ConjNormAlgStr;
reserve a,a1,a2,b,b1,b2 for Element of N;
reserve c,c1,c2 for Element of Cayley-Dickson(N);

theorem Th35:
  <%<%<%0.N_Real,1.N_Real%>,<%0.N_Real,0.N_Real%>%>,
  <%<%0.N_Real,0.N_Real%>,<%0.N_Real,0.N_Real%>%>%>
  *
( <%<%<%0.N_Real,0.N_Real%>,<%1.N_Real,0.N_Real%>%>,
  <%<%0.N_Real,0.N_Real%>,<%0.N_Real,0.N_Real%>%>%>
  *
  <%<%<%0.N_Real,0.N_Real%>,<%0.N_Real,0.N_Real%>%>,
  <%<%0.N_Real,1.N_Real%>,<%0.N_Real,0.N_Real%>%>%> )
  =
  <%<%<%0.N_Real,0.N_Real%>,<%0.N_Real,0.N_Real%>%>,
  <%<%0.N_Real,0.N_Real%>,<%1.N_Real,0.N_Real%>%>%>
  proof
    set a1 = ZZJZ*ZZZZ-ZJZZ*'*ZZZZ, b1 = ZJZZ*ZZJZ+ZZZZ*ZZZZ*';
A1: ZJ*ZJ*' = <%z*z-(-j)*'*j,(-j)*z+j*z*'%> by Lm3,Def9;
A2: JZ*ZJ = <%j*z-j*'*z,j*j+z*z*'%> by Def9;
    b1 = <%ZJ*ZZ-JZ*'*ZZ,JZ*ZJ+ZZ*ZZ*'%> by Def9
    .= <%ZZ,JZ*ZJ%>;
    then
A3: b1*ZJZZ = <%ZZ*ZJ-ZZ*'*ZJ,ZZ*ZZ+ZJ*ZJ*'%> by A2,Def9
    .= <%ZZ,JZ%> by A1;
    <%ZZJZ,ZZZZ%> * <%ZZZZ,ZJZZ%> = <%a1,b1%> by Def9;
    hence <%ZJZZ,ZZZZ%> * (<%ZZJZ,ZZZZ%> * <%ZZZZ,ZJZZ%>)
    = <%ZJZZ*a1-b1*'*ZZZZ,b1*ZJZZ+ZZZZ*a1*'%> by Def9
    .= <%ZZZZ,b1*ZJZZ%>
    .= <%ZZZZ,ZZJZ%> by A3;
  end;
