then A5: 0
then consider n1 such that
A6: for
m st n1<=m holds abs(seq.m-g1)
consider n2 such that
A7: for m st n2<=m holds abs(seq'.m-g2)
take k=n1+n2;
let m such that
A8: k<=m;
n1<=n1+n2 by NAT_1:37;
then n1<=m by A8,XREAL_1:2;
then A9: abs(seq.m-g1)
n2<=k by NAT_1:37;
then n2<=m by A8,XREAL_1:2;
then abs(seq'.m-g2)
then A10: abs(seq.m-g1)+abs(seq'.m-g2)
A11: abs((seq+seq').m-g)=abs(seq.m+seq'.m-(g1+g2)) by SEQ_1:11
.=abs(seq.m-g1+(seq'.m-g2));
abs(seq.m-g1+(seq'.m-g2))<=abs(seq.m-g1)+abs(seq'.m-g2) by COMPLEX1:142;
hence abs((seq+seq').m-g)
end;