This may be a weird question but for a BJT, we have
\$I_S\$ is the reverse saturation current and \$I_C\$ is the collector current which flows through all three parts of the BJT.
Now my question is, \$I_S\$ should also flow through all three parts of the BJT, so can't we just add/subtract \$I_S\$ from \$I_C\$ via KCL? Is that where the -1 comes from?.