 reserve Rseq, Rseq1, Rseq2 for Function of [:NAT,NAT:],REAL;

theorem
   Partial_Sums_in_cod2 Rseq = ~(Partial_Sums_in_cod1 ~Rseq)
 & Partial_Sums_in_cod2 ~Rseq = ~(Partial_Sums_in_cod1 Rseq)
 & ~(Partial_Sums_in_cod2 Rseq) = Partial_Sums_in_cod1 ~Rseq
 & ~(Partial_Sums_in_cod2 ~Rseq) = Partial_Sums_in_cod1 Rseq
proof
   now let n,m be Element of NAT;
    (Partial_Sums_in_cod2 Rseq).(n,m)
      = (Partial_Sums_in_cod1 ~Rseq).(m,n) by Tr2;
    hence (Partial_Sums_in_cod2 Rseq).(n,m)
      = ~(Partial_Sums_in_cod1 ~Rseq).(n,m) by FUNCT_4:def 8;
   end;
   hence Partial_Sums_in_cod2 Rseq = ~(Partial_Sums_in_cod1 ~Rseq)
   by BINOP_1:2;
   now let n,m be Element of NAT;
    (Partial_Sums_in_cod2 ~Rseq).(n,m)
      = (Partial_Sums_in_cod1 Rseq).(m,n) by Tr2;
    hence (Partial_Sums_in_cod2(~Rseq)).(n,m)
      = (~Partial_Sums_in_cod1(Rseq)).(n,m) by FUNCT_4:def 8;
   end;
   hence
a1:   Partial_Sums_in_cod2(~Rseq) = ~Partial_Sums_in_cod1(Rseq) by BINOP_1:2;
   now let n,m be Element of NAT;
    (~Partial_Sums_in_cod2(Rseq)).(n,m)
     = (Partial_Sums_in_cod2(Rseq)).(m,n) by FUNCT_4:def 8;
    hence (~Partial_Sums_in_cod2(Rseq)).(n,m)
     = (Partial_Sums_in_cod1(~Rseq)).(n,m) by Tr2;
   end;
   hence ~Partial_Sums_in_cod2(Rseq)
    = Partial_Sums_in_cod1(~Rseq) by BINOP_1:2;
   now let n,m be Element of NAT;
    (~Partial_Sums_in_cod2(~Rseq)).(n,m)
      = (~Partial_Sums_in_cod1(Rseq)).(m,n) by a1,FUNCT_4:def 8;
    hence (~Partial_Sums_in_cod2(~Rseq)).(n,m)
      = (Partial_Sums_in_cod1(Rseq)).(n,m) by FUNCT_4:def 8;
   end;
   hence ~Partial_Sums_in_cod2 ~Rseq =Partial_Sums_in_cod1 Rseq by BINOP_1:2;
end;
