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

theorem
  Rseq is non-decreasing implies
    (for m being Element of NAT holds ProjMap1(Rseq,m) is non-decreasing) &
    (for n being Element of NAT holds ProjMap2(Rseq,n) is non-decreasing)
proof
   assume a1: Rseq is non-decreasing;
   hereby let m be Element of NAT;
    now let n1,n2 be Nat;
a0:  n1 in NAT & n2 in NAT by ORDINAL1:def 12;
     assume n1<=n2; then
     Rseq.(m,n1) <= Rseq.(m,n2) by a1; then
     ProjMap1(Rseq,m).n1 <= Rseq.(m,n2) by a0,MESFUNC9:def 6;
     hence ProjMap1(Rseq,m).n1 <= ProjMap1(Rseq,m).n2 by a0,MESFUNC9:def 6;
    end;
    hence ProjMap1(Rseq,m) is non-decreasing by SEQM_3:6;
   end;
   hereby let m be Element of NAT;
    now let n1,n2 be Nat;
a2:  n1 in NAT & n2 in NAT by ORDINAL1:def 12;
     assume n1<=n2; then
     Rseq.(n1,m) <= Rseq.(n2,m) by a1; then
     ProjMap2(Rseq,m).n1 <= Rseq.(n2,m) by a2,MESFUNC9:def 7;
     hence ProjMap2(Rseq,m).n1 <= ProjMap2(Rseq,m).n2 by a2,MESFUNC9:def 7;
    end;
    hence ProjMap2(Rseq,m) is non-decreasing by SEQM_3:6;
   end;
end;
