reserve n for Nat;

theorem Th6:
  for i,j,m,n be Nat holds i+j = m+n & i <= m & j <= n implies i = m
proof
  let i,j,m,n be Nat;
  assume that
A1: i+j = m+n and
A2: i <= m and
A3: j <= n;
  consider k be Nat such that
A4: m = i + k by A2,NAT_1:10;
  consider l be Nat such that
A5: n = j + l by A3,NAT_1:10;
  j + (l+k) = j+0 by A1,A4,A5;
  then k = 0;
  hence thesis by A4;
end;
